Yes, ethnic profiling against the Jews took place at least in 1970s-1980s, and, IIRC, at other times. The favored ethnicity was Russian, and other Slavic nations were acceptable as second-tier, but also favored.

Note also the ethnic Jews in other countries with weaker antisemitism than that in the USSR made up higher proportion of the political (and scientific) elite. As an example, take the lists of prestigious scientific award winners (Nobel prize, Fields medal), and calculate the fraction of ethnic Jews among those. I leave this an an exercise to the reader. I expect about this fraction of Jews in the ruling elite of the USSR, but AFAIR it was much, much lower. Feel free to search the interwebs for USSR antisemitism, to find a few good references re discrimination of Jews in favor of Russians and other Slavs at the state level.

REFERENCES:

I personally spoke in 1980s with a high-ranking Jewish scientist N., who had been working in the defense sector in USSR. N. explained to me that he had been explicitly told by a very high ranking Soviet official that they would have appointed N. to the minister or assistant minister position ("ministr" or "zamestitel' ministra" in Russian) had he not been a Jew. This was expressed with regret by that official, who was sympathetic to N., but who could not violate the orders from above.

I do not reveal the name of N. for fear of repressions that might be initiated against his family that currently lives on the territory controlled by Putin's regime. This was personal communication with me, and I cannot produce any written proof.

You can try googling and translating "этнический состав советской номенклатуры".

I have stumbled on an article which says:

В государстве, где «ядром политической системы» была единственная правительственная партия, элиту составляли в первую очередь лица, входившие в Политбюро (в 1952 – 1964 гг. оно называлось Президиумом), Оргбюро и Секретариат ЦК. Каков был их национальный состав?... Больше всего в этих органах было русских - 147 человек, что составляло 64 %; украинцев было 18 человек (7,9 %), евреев – 12 человек (5,2 %), белорусов и латышей – по 8 человек (по 3,5 %), грузин – 6 человек (2,6 %), армян и узбеков – по 4 человека (по 1,7 %), азербайджанцев было трое (1,3 %), казахов, киргиз, молдаван, немцев, поляков и эстонцев – по 2 человека (по 0,9 %), болгары, литовцы, осетины, таджики, татары, туркмены и финны – по одному человеку (по 0,4 %). Один человек () остался за рамками этих подсчетов (отец у него был молдаван, а мать – русская).

Members of Politburo, Orgburo and communist party secretariat were:

• Russians: 64%
• Ukrainians: 8%
• Jewish: 5%
• Belorussians and Latvians: 3.5% each
• Georgians: 2.5%

...long tail follows...

It's hard to say whether the Russians sitting in the Politburo were representative of general Russian population or not, but it doesn't seem there's significantly more of them with regards to overall ethnic composition of the USSR, and you can spot some privileged minorities such as Georgians and Latvians (e.g. the second-to-last minister of interior of USSR, Boris Pugo, was Latvian, but born in Russia, in communist "Latvian Riflemen" family).

Historical Background

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.

Ethnic Policy

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 ;-).

Cadre Policy

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 protests.

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 Politburo, the data will not be very representative - too small data set. There will be mostly Russians (and some Ukrainians and Belorussians), sprinkled with Latvians, Georgians, Kazakhs, Jews, Armenians &c 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.

Council of Ministers of the Soviet Union

If we want to look at the official national government, we can rely on several lists for various dates, e.g., 1979-04-19: 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 decided).

Central Committee of the Communist Party of the Soviet Union

For some dates there are full lists, but not everyone on the list has their own wiki page.

E.g., 1952: 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.

See also

• Советская номенклатура.
• How were USSR oblasts and cities governed?

Postscriptum

All 3 answers to this questions have several downvotes. I think this is a sad illustration of politicization of the site.