Yes, as the Russian Federation is legally the successor state of the USSR and assumes all obligations, roles, debts, etc. of the former entity. Interestingly, Ukraine also claims that it's the successor state of the USSR. Obviously, this claim has not be recognized by the UN as the Russian Federation has the previously owned permanent seat on the security council.
This situation is not new. A similar situation happened with China, as China was undergoing a civil war in prior to WWII and still was during the conclusion of the war when the UN was founded. The country known currently known as Taiwan was recognized as "China" (i.e. the Republic of China). The government of the Republic of China lost the civil war and the People's Republic of China was founded while the former government retreated to the island of Taiwan. Both entities claimed to be the legal successor state of "China" but in 1971, Taiwan was no longer recognized and the People's Republic of China was formally recognized as the successor state of the Republic of China. In fact, Taiwan is not recognized at all by the UN, so the situation is slightly different from the USSR situation.
But like the People's Republic of China, the Russian federation assumes all responsibility, obligation, etc. of its predecessor. A clear example of this for China is the 99 year lease of the Hong Kong territory, which was made by the Qing dynasty and fulfilled by the People's Republic of China. There are numerous example for the Russian Federation; e.g. the precursor to the international space station program, Mir, had its former Soviet obligations fulfilled by the Russian Federation.