You have to end the Cold War and somehow make the Soviet Union a relevant and viable competitor in an area that isn't related to warfare or revolution. As others have mentioned, the USSR was slowly strangled to death by the Western bloc's superior economic system. There two factors to this: one external, the other internal, and both results of the Cold War framework.
Problems
The external factor is that of the strength levied against the USSR from the West and China. The Soviet Union was seen as an aggressive empire keen on expanding its borders through violence and revolutionary subversion. This view is vindicated by Soviet (and in fact modern Russian) behavior vis-a-vis bordering states, most notably the shackling of the Eastern Bloc via the Warsaw Pact, and subversive activities in the 3rd World.
But all this, rather than being the reflection of Soviet strength, is actually a reaction to its own inherent weaknesses. Geographically, the USSR was no match for the West and China, which between them controlled pretty much the whole of the worthwhile world. Additionally, the Soviets were materially and demographically weakened by World War 2, further undermining their competitive value.
The internal factor is the way the Soviet leadership saw itself, its people, and the outside world. Being communists, Soviet leaders, at least on paper, felt it necessary to adhere to a few tenets. First and foremost, in 1960 they still believed in the vitality of their own economic system as an alternative to the capitalist one. Secondly, they also found it imperative to maintain the Marxist revolutionary ideals, which impacted their domestic and international condition.
While the practice of these ideals on the international stage could and was manifested as the "fraternal" relations with Eastern European countries and 3rd world movements, it also fit into the defensive mindset of the USSR leadership. On the internal front, the Soviet leadership ruled through the Communist Party and thus felt the need to keep the populace faithful to communism as such.
The result of these two factors in concert meant that the Soviet Union in 1960 was firstly stuck in an unenviable defensive position, and secondly that its leaders somehow believed or felt pressured to believe that this situation could be remedied through maintenance of Communist ideals and the economic system.
Solutions
Delivering the USSR from the trap of these two factors—that is, making the USSR not a categorical enemy of the West, and freeing its leaders from the ossified thinking of Marxism—is difficult but it probably can be done with the right luck and skill. First you have to make the USSR recognize that it is never going to be able to maintain military superiority or even parity with the West. Second you have to have the Soviets abandon Marxism in all but name, similar to what China did.
The first goal is very dangerous but not so hard. Given the PoD is 1960 or later, we can have the USSR suffer some sort of horrific setback, Cuba being a possibility. Let's say the US moves to remove Castro (and the Soviet weapons) in during the Crisis; I personally doubt the Soviets would let themselves get destroyed over Cuba. This embarrassment would not only remove Khrushchev from power; it could firmly show the USSR that military competition with the West is neither sane nor winnable.
Following this, the USSR leadership itself debates the utility of an economic system that has produced so much inefficiency and weakness. The USSR might give up the ideal of world revolution completely, allowing Eastern Europe to Finlandize (as a pilot project) in return for remaining under the Soviet military system. A Tiananmen-style protest for democracy is crushed brutally, and is then ignored as the Soviets allow Eastern Europe to function as "special economic zones." The USSR props the newly-capitalist nations such as East Germany, Poland, or Czechoslovakia up with cheap oil and raw materials, allowing them to compete with superior Western products.
Later, around 1970, a military confrontation with China turns into a regional war that the USSR wins, restoring faith in the leadership, the strength of the military establishment, and possibly retarding the development of China as a major power.
Between 1970 and 1990, the USSR emphasizes "peaceful development." Political reforms are off the table, at least not for the Soviet Union itself; the main thing is to make money. The West, seeing the economic reforms in Eastern Europe and the clobbering of China at the height of its own ideological madness, is satisfied and no longer considers the Soviets to be an existential threat. NATO remains, as does the Warsaw Pact, but they are simply there to "keep the peace"—ideology has taken a strictly back seat to business.
By TTL's 1980, the USSR survives off of oil and raw mineral exports, but the Soviet elite also wants in on the free market action, so they start using the state-run enterprises in collusion with first Eastern European, then homegrown private firms to engage in all kinds of corrupt business. The more politically open Eastern Europeans protest, but the threat of the Warsaw Pact and their economic comfort precludes any serious unrest.
By 1990, Moscow, Leningrad, and Kiev are major centers of Soviet market investment, and the rest of the country languishes—they are not profitable. Soviet citizens emigrate from stagnating Stalin-era towns to the modern cities and Eastern Europe, which is forced to accept the influx. Ideologically, Communism is all but dead; what has replaced it is Great Russian nationalism under a crude mask of pan-Slavic brotherhood. Religion is still heavily influenced by the state but there is no longer any official stigma against faith.
In the 1990s things start to get dicey in Russia other parts of the USSR. Intense corruption backed by the Party structure causes popular unrest; while criticizing the Party itself is taboo, it is common for officials and state services to be demonized in the eyes of the public. Millions of less-fortunate Soviets in the left-behind Siberian rustbelt, as well as the actual political dissidents who couldn't be either bought off or eliminated, are a growing threat to the leadership as it becomes clear that the authoritarian state-run capitalist system has its own problems (mostly corruption and frequent lawlessness).
Circa 1995 there is some sort of crisis (could be an alternate Chernobyl or some sort of political fight, think Xi Jinping versus the corrupt officials of OTL China) that undermines the one-party system of the USSR. At the same time, ethnic tensions arise, though they are less pronounced or dangerous since the USSR has been spending about two decades pushing a very pan-Slavic platform that did not exist IOTL.
As the combination of these factors threatens collapse, a coalition of Party "reformers" and the Soviet military and police establishment launch a coup that overthrows the Communist Party and co-opts its authoritarian structure while crushing any hope of independence for uppity regions. This authoritarian monopoly is broken up to allow for multiparty politics while preserving the remnants of the CPSU as a default "big tent" party that panders to "all true Slavs and Russians". While true opposition parties now exist, the former CPSU is still indisputably the largest and leads the nation. Its platform is basically a nationalist welfare state. However, it will never hold the same political monopoly that the CPSU enjoyed; this fact, coupled with popular demand, leads to a vastly more transparent Soviet government. As a result, it is less wasteful, less prone to brazenly violating Soviet law, and longer-lived.