The only way I can see Kosygin at the top is similar to how Gorbachev did so- the more hardline sections of the party see several of their leaders die off in quick succession, leaving little contest to his ascension. Problem is, Suslov dies in 82, Pelshe in 83, and Andropov in 84, while Kosygin died in 80.
now, suslov dying earlier is quite possible, as he was rather sickly, and may be necessary for this scenario. So Brezhnev is assassinated in 1969, and Pelshe succeeds him, with Suslov acting as the power behind the throne. Andropov is sidelined by Pelshe with tacit approval from Suslov, to remove a potential rival. Kosygin takes advantage of the chaos to reestablish and strengthen himself, and Suslov/Pelshe for whatever reason do not stop him. Suslov dies in 1976 as a result of his heart trouble, (maybe earlier would be better; say around '71 or '72). Pelshe himself had lung cancer, although I cant remember when he got it, so maybe stress from the aftermath of Suslov;s death eventually leads to his own demise a year or two later. Brezhnev, one of Kosygin's main opponents is now gone, and with Suslov and his protegee's either dead or out of the way, Kosygin might rise to power in the vacumn.
Now, the problem is, especially if we use Suslov's health to get the ball rolling, we leave Kosygin with a truncated term unless we can find a way to extend his life (two or three years may be interesting, but will not have major ramifications). we are probably going to have to move up Suslov's death to make this work (more time for Kosygin and Pelshe is less established), which causes complications of it's own (for one, Pelshe's own health).
Now, maybe if somebody else was in Brezhnev's car (anybody know who the occupants were IOTL), that could give us more butterflies to work with.