Logically, in every war ever fought in Asia, there must have been a winner. If not overly pacifist in its reading, or expecting all too many stalemates.
This dictum is a jokingly formulated advice, not anything remotely like a historical summary of events.
An unstated prerequisite assumption in the question title is euro-centric: obviously from Sumerians onwards many Asian powers fought and won wars in Asia. Mongols won most wars in Asia, before also visiting Europe.
If the question is then about non-Asian powers winning in Asia: then it is also obviously, the Russian Empire won quite a few wars going east (example: Russian conquest of Siberia (1580 – late 17th century)), as did the English when colonising large parts of it, most notably India.
This is arguably 'an old tradition', since the time of the Macedons and ancient Rome. Alexander III left Europe, and except for a small detour to march into Egypt, went to war with Persia mainly in Asia until he reached the Indus. (hat tip to @Jan from comments) Rome started to win a lot of wars in Asia after the battle of Magnesia.
Later, the British Company Raj won for example the Second Anglo-Maratha War (1803–1805), Third Anglo-Maratha War (1817–1818), Indian Rebellion of 1857, or the First Anglo-Sikh War (1845–1846) and the Second Anglo-Sikh War (1848–1849), to name just a few.
So, given this map about Colonialism, most land parts of Asia and quite a bit of waters were conquered sucessfully by European powers, plus a bit of American 'influence':
And that picture leaves out those uncoloured specs over which or 'with whom' wars were fought, and often won, but which did not result in permanent rule over the territories.
Examples for that include wars with China (the Opium Wars) or Afghanistan. The latter of course being famous for a lot a pyrrhic victories, starting with the First Anglo-Afghan War, albeit the British even achieved their objectives for quite some time with the Second Anglo-Afghan War.
As a 'non-Asian power', the US alone could account for victories in Asia in the Philippine–American War, Moro Rebellion, Boxer Rebellion, and the little episode called World War Two. Even the Korean War wasn't 'lost', strictly speaking. Also, the Tanker War, Gulf War, "Enforcement of Iraqi no-fly zones", and the "Operation Ocean Shield" are commonly counted as successful for the American side.
In the case of quoting the 1987 movie or the 1973 novel this adage is just a joke.
The origin of this phrase is from the aftermath of the US experience in the Korean war and gained much traction from 1964 onwards for perhaps applying this 'lesson learned' also to decisions the US made regarding Vietnam. The early usage for this means '[The US better] never again fight a ground war in Asia' as a general concern and prevalent feeling, not dominating but significant, within even the US military, albeit most often anonymously. Example:
That the U. S. not undertake a prolonged land war in Asia's vastness where China's man power might be effective if the war bogged down. […]
No responsible military leader would choose to commit U.S. troops to a ground war in Asia. No one wants to grind away American manpower in a seemingly endless war against numerically superior forces deep inside China. The U.S. objective, however, is not to fight for the heartland of China, but to prevent the spread of communist control over Asia.
— Lloyd Norman: "No More Koreas", Army (United States Army Combat Forces Journal), Volume 15, Part 2, May 1965. (p31, gBooks)
Or more explicitly, slightly later:
A number of army officers had been skeptical and remembered the lesson of Korea to never get involved in a land war in Asia; however, […]
— Erwin C. Hargrove: "The Power of the Modern Presidency", Temple University Press: Philadelphia, 1974. (p126)