Continuity is the key word when you think about Russia.
There have been major upheavals in Russian history in the 20th century (the Bolshevik revolution in 1917 - and the ensuing civil war!, and the relatively peaceful collapse of the USSR in 1991) which temporarily took Russia away from its "normal" path, but it returned to it within ~10 years.
Russia's national ("state", not "ethnic") mentality is the inevitable necessity of expansion to improve security (see Diplomacy by Henry Kissinger). Russia is "peaceful" while it is digesting the newly acquired lands (the aggression is directed against the remnants of the independence movements), and aggressive, when it feels that the recently digested territories are threatened by the still-independent neighbors.
The ethnic policy in Russia (except for the brief aberrations after the 1917 and 1991 upheavals, when the state is weak and fears national break up) has always been the same: "Russification" of all non-Russians. It was a very persistent but somewhat inconsistent (not pushed equally hard everywhere) policy, which was clothed differently depending on the way the national ideology was currently formulated.
E.g., during the Soviet epoch it was called "national in form, socialist in content", which meant that local languages could be used to spread the official ideology.
Use of local languages beyond secondary education was actively discouraged.
(Similar to medieval universities teaching in Latin ;-).
Now for the question asked: the key attribute in a person aspiring for a political position has always been loyalty to the ideology, i.e., Russian Imperial world view in the current incarnation (e.g., 1920-ies - world revolution lead by Russia, 1930-ies - "socialism in one country", before 1917 and now - "Russian world").
Thus, Russians were always preferred, but not because of "racism", but because a Russian is more likely to support Russian expansionism and Russification of non-Russians. Ukrainians and Belorussians were second best because they were ethnically and linguistically close. Jews could never be trusted, like everywhere else - see the excellent other answer.
Government Composition as a matter of Policy
One standard aspect of the Soviet system is that the power center was the Party, not the official
Soviet administration. All the actual decisions for a region ("область") were made by the regional
party bureau, chaired by the First Secretary of the regional party organization. The "Executive
Committee Chairman" (председатель исполкома) was merely a bureau member, outranked not only by the
First Secretary, but also by his 2 deputies (Second and Third Secretaries).
In the Russian regions, the power ranking was as one would expect: the First Secretary is the boss,
the Second is his deputy &c. and they were always Russian (culturally if not ethnically, e.g., it
might be a Ukrainian who speaks little Ukrainian language).
However, in ethnic regions the First and Third Secretaries were always "locals" (e.g., they were
Uzbeks in Uzbekistan) and the Second was always a Russian, and the true power center was the Second
Secretary, not the First one.
When Kazakh Kunaev was replaced as the First
Secretary of the Communist Party of Kazakhstan by Russian
Kolbin in December 1986, there were violent
Government Composition as a matter of Statistics
When talking about ethnic composition of the government, one has to decide the level of cut off.
E.g., if we limit ourselves to
the data will not be very representative - too small data set.
There will be mostly Russians (and some Ukrainians and Belorussians), sprinkled with
The other answer collected the stats.
The over-representation of Jews is due to the post-upheaval 1920-ies.
If we want to look at the nomenklatura as a whole, the
statistics is not available because the lists of members thereof were always classified.
If we want to look at the official national government, we can rely on several lists for various
out of 93 names, only a handful can be suspected of non-Slavic heritage.
Out of those, there are 2 Jews, and one Azeri, Bashkir, and German each.
Two more "suspects" turned out to be half Russian and half non-Slavic, which, for all intents and
purposes, means Russian (ideologically, not "racially", as explained above).
The same list for 1962-04-25:
56 names, 2 Armenians, 1 Jew.
The same list for 1946-03-19: 56 names, 4 Jews, 2 Georgians, 2 Armenians.
This is not very interesting though, as the Council of Ministers was a technical organ, not a
political one (i.e., it did not decide policy, by merely implemented whatever the Party has already
For some dates there are full lists, but not everyone on the list has their own wiki page.
125 full members (2 Azeri, 8 Central Asian, 4 Georgians, 2 Armenians, 3 Jews, 2 Baltic, 1 Finn),
111 candidates (1 Armenian, 3 Baltic, 2 Georgian, 1 Armenian, 1 Central Asian, 1 American(???!!!)).
Again, this is based on names only, please take it with a grain of salt.
Basically, the rest looked very Slavic, but these did not.
All 3 answers to this questions have several downvotes.
I think this is a sad illustration of politicization of the site.