As this regards an obscure period of European history, all we can hope is speculate within the bounds of reason.
It might prove useful to consider the larger issue of the Eastern Romance substratum. As Oldcat answer points out, the insular position of the Romanian language might be seen as the result of a survival as part of a larger Latin region.
From this perspective, the answer to the question comes from clarifying that the exclusive "Roman-ness" of Romanians in the Balkans in not their doing, but rather the result of larger Slavic influence and presence south of the Danube. Slavs (that is: Slavic-speakers) were also present on the territory of Romania but probably in somewhat smaller numbers.
Why were there more Slavs south of the Danube then north of it? And even so, weren't even more Romans south of the Danube, where the Roman presence was older, than Slavs?
The Slavs were great agriculturalists. Most important agrarian terms in Romanian are of Slavic origin and even now Romanian of Wallachia see Bulgarians as the greatest at cultivating vegetables. Romanians see themselves as a people of land-workers, but also of shepherds. There was a strong trend in Romanian historiography and culture linking Romanian continuity and Latin identity to pastoral activity and culture. Even exaggerated, that may stand for something. Large parts of the now mainly agricultural land of southern Vallachia was deep forest until recently. Possibly the Slavs were looking for open land and that was more available south of the Danube. Dacia was maybe not such a hospitable land compared to the riches closer to the heart of the empire, and it was surely more exposed to new invasions.
Possibly the Latin language south of Danube was more present in cities than in villages, while the Dacian provinces north of the Danube were less urbanized. Cities were the first victims of invasions and the urban Roman population had possibly better means to flee the invasions than the people of the lesser urbanized Dacia. For some reason the Latin speakers north of the Danube must have found themselves without urban structures, in the impossibility of fleeing, or in a situation where they could survive without fleeing (adapted to pastoral mountainous regions). The presence of Latin-speaking populations outside Romania in the Balkans has diminished with the assimilationist modern nation-states, but their traditions were mainly pastoral.
The assimilation of one language into another needs certain factors. The superstratum language has to provide certain economical and political advantages in order to replace the substratum language. (As Latin did when replacing many European languages). As the Roman state receded and was replaced by the newcomers, even if the Slavs ruled at some point in present Romania, political stability and unification may have been less present than in the Slav-speaking kingdoms south of the Danube, and possibly only there the necessary incentives to the adoption of Slavic may have been present. Themselves conquered by the Bulgars, the Slavs of Bulgaria gained political structures that were surely more developed than the ones north of the Danube: the first relatively stable Romanian polities in Wallachia date from 13th century; the Bulgars created their first empire in the Balkans in 681 and in 879 adopted Christianity; they gradually adopted the Slavic language which thus became the language of their empire, important enough to impose the creation of the Cyrillic alphabet. Meanwhile, north of the Danube new invaders were ruling, the Pechenegs and Cumans. But following the logic above, they could not succeed where the Slavs had failed: the Latin language there was not replaced. The notorious lack of political stability and of statehood north of the Danube was seen as a very improbable environment for a Latin language to survive. But the opposite might be true: absence of non-Latin political structures and stability in the Dacian/Romanian area may have contributed to the conservation of the Neo-Latin substratum language spoken by rural and pastoral mountainous people. When invasions ceased and the first states appeared north of the Danube it was too late for the language to change.
(Two powers might have played a decisive role in this obscure period of history: the Bulgarian empires and the Hungarian kingdom. The tzars of Bulgaria possibly extended their realm north of the Danube, which may raise the possibility of some movement of population. But the north of the Danube seems a territory of great instability and invasions all the way, and there is no proof of great effort from Bulgarian tzardoms of creating a stable political state north of the Danube, in the absence of which it remains greatly improbable that people could have had any incentives of moving northwards in the path of danger. There is on the other hand proof of Hungarian effort in this sense, which culminated in the creation of the two principalities at the beginning of the 14th century, after the end of the last great Mongol invasion and the stabilization of the Golden Horde north of the Black Sea. This Hungarian intervention involves no population from south of Danube, but most probably local and Transylvanian.)
On the other hand, one shouldn't forget that the question is about the survival of the language, not of the people. The present Romanians were not isolated from their neighbors. Romanian is a Romance language but the only one with significant Slavic influences. These influences were much more present in the vocabulary at the beginning of the 19th century than they are now (especially in written language; most of the population was then illiterate, the political and religious elites used Slavonic; Cyrillic alphabet was used until mid-19th century).
(And the influences came both ways. I am not expert in Slavic languages, but "месец" ("mesetz") means both "moon" and "month" in Serbian, while in Bulgarian it only means "month"; the Bulgarian word for "moon" is "луна" ("luna"), like in Romanian (and Latin). On the other hand, just like the Serbs (and unlike the Bulgarians, English and... Latin), albeit using a Latin word, Romanians have one single word for "month" and "moon".)
Also, it is interesting to consider that languages of the Balkan cultural and geographic area share certain common features that are independent of their Indo-European or Latin ancestry: the presence of a linguistic Balkanic continuity or unity - the Balkan sprachbund.
The 19th century Romanian national independence, unification and modernization coincided with a (re)discovery of the Latin identity, which gradually became the main marker of national identity. The French influence was enormous between 1850 and 1900, both culturally and politically (pushing against the receding Turkish and the increasing Russian hegemony), and contributed to a re-latinization of the language - both directly (through the adoption of French terms) and indirectly (through the adoption of Western and Latin referents of cultural identity). Only then Romanians started to view themselves as closer to Italians and French then to Bulgarians and Serbs, a trend that has its part of artificiality but is again very dominant (after the fall of communism and the renewal of pro-Western influence in that part of Europe).
UPDATE in relation to an answer that tries to dismiss the topic of the question arguing that Romanians are not the main Latin/Roman cultural phenomenon of the Balkan area, and that the same question should consider other peoples, like the Croats:
The argument considers as "Latin" or "Roman" other facts than the neo-Latin language:
The Catholic confession. That is considered to be a marker of Roman identity because its center is Rome and that Latin was used as cultural and liturgic language in Catholic countries, unlike in the Orthodox Romania.
The anthropological/genetic identity, based on which other peoples (e.g. Croats) should be considered as closer to the Roman/Latin identity.
But the Romanians are discussed here as "Latin" mainly in relation to their neo-Latin language. Using the word "Latin" in relation to the Catholic confession (because the Popes lived mostly in Rome and were mostly Italians themselves) is using the same word with a different meaning. Neo-latin identity and language is the topic here, not the use of the classical, liturgical or scientific Latin language. The question above could be read like this: why is there a (neo)Latin culture in Eastern Europe? Answering that Western Europe is by definition "more Latin" than anything that can happen in the Eastern Europe (and even that a Balkan country geographically more to the West is intrinsically "more Latin") is missing the point completely.
To give an example of how a different use of the word "Latin" may completely alienate the topic: while the Western kingship and nobility order is of Germanic origin, the rulers of Orthodox Christianity, namely the Bulgarian and Serbian kings, the Russian tzars and the Romanian voïevods were all absolute rulers within a system taken from the Byzantine emperor, which was that of the Roman Empire itself. That this system ended up being called "tzarism" instead of "Roman" is due to the fact that Russia has kept it until recently (in a modified, dynastic form, closer to Western absolutism). But, far from being related to some oriental tyranny, it is also a Roman heritage (and the term "tzar" comes from Caesar). -- But all this has nothing to do with the question.
As for the non-cultural (that is biological/genetic) Roman-ness (the author of the answer even mentions an "ethno-racial connection" between Croats and Italians), no matter the incongruity of a such non-cultural element when the topic is cultures and history, it may sound plausible that some Balkan people other than Romanians may be "more Latin" (by which the author probably means more sun-bathing) because they live on the shore of the Mediterranean. But facts are stubborn. Taking a look at European genetics (for example here or here), and namely at the map below, it is easy to see that the Red+Green combination on that map (Celtic+Greco-Roman markers) which is defining Italy is also very important in Romania. But that proves nothing as to the "Latin" character discussed here, for that "Italian" combination is also significant in Austria, Hungary, Albania, and even Greece and Turkey. Therefore, the language remains the sole meaningful marker on the above topic.
The same answer, while pretending to clarify the issue, brings further confusion by presenting as a significant point the fact that
Wallachia or "Land of the Vlachs", is actually in Southern Romania and
is not representative of the entire country
The South of present Romania became a stathood known as Wallachia, which in Romanian was called Muntenia (munte means mountain) or Ţara Românească ("Romanian country"), but it is not at all true that only people of the southern region were called Vlach/Wallachian. The transhumant pastoralist Vlachs that reached Moravia (Moravian Wallachia) were called by the same name. The problem is different: Romanians became called Vallachian or Vlach by their Hungarian, Slavic and Germanic neighbors, a term which at the beginning simply means Roman or Italian (possibly related to terms like Welsh and Walloon). Especially Hungarians (who dominated initially all Romanian principalities and were instrumental in the creation of Moldavia as buffer-state against the last Mongol invasion) used the term to name the Romanians, first in Transilvania, then in Wallachia (therefore called Hungro-Wallachia) and Moldavia (at times called Moldo-Wallachia or Black Wallachia). It is the term Wallachian or Vlach (the latter only in modern times used specifically to name the neo-Latin speakers south of the Danube) which gave the name of the Principality of Wallachia, not the other way around.