It depends on the period of time we are talking about, but the basic Soviet system (so far as we know) worked like this (from a paper I wrote but never did anything with):

In the Soviet Union, authority over nuclear forces, both their production and use, were in principle centralized within the Politburo of the Communist Party Central Community. Contrary to the American system, in many ways this approach was more centralized, a top-down approach to governance that saw fewer of the typically American distinctions between legislative and executive branches, between civilian and military authorities. In practice, however, many decisions took place at a level “below” the ultimate political authority, at the level of the Ministry of Defense and General Staff, i.e., the military itself. Depending on the nature of the defense matter, it would be handled by various sub-levels of bureaucracy. Where the Soviet system appears to have differed significantly from the American system, however, is how authorization of nuclear use would take place. In the Soviet imagination about nuclear war, it comes as an overwhelming counterforce attack by the United States (and NATO), an attack that would seek to destroy its capacity to respond in turn. As such, it is geared towards an extremely rapid, obliterating response, one in which the amount of decision-making time is extremely short, despite extensive early-warning computer and sensor systems. Much of the doctrine revolves around the personage of a Supreme Commander of the armed forces, a military officer given the authority to coordinate and launch a retaliatory strike by the political apparatus in scenarios where a nuclear attack might be judged to be plausible. Once appointed, the Supreme Commander would not require permission from the Politburo to use strategic forces in retaliation; instead, their nuclear authority was automatic if the early-warning system had sent a “missile attack” signal.

The information comes from Pavel Podvig, Russian Strategic Nuclear Forces (Cambridge, Mass.: MIT Press, 2001), 39-42 and 59-66.

So the short answer is: the Politburo, but that only turns over executive authority to the Supreme Commander of the Armed Forces. There is no singular, unilateral use equivalent in the Soviet system. (Nor is there in the modern Russian system, which gives partial authority to three people, of whom two must concur.)