Both sides. The Cuban Missile Crisis is an example of an iterated Chicken Game [1] - it goes something like this:

If both sides fail to launch nuclear weapons, the status quo is upheld, and so the score is [0, 0]. If both sides play this strategy, they get to keep playing the game (i.e. they stay alive).

If both sides launch nuclear weapons, MAD [2] ensues and the score is [-10, -10]. Game over.

If only one side launches, then they will, at least at the end of the first round gain a strategic success, making the score [+1, -1]. Nevertheless, as, going into this iterated (repeated more than once) game we know that the response for round two if one side launches is annihilatory retaliation. Both sides know that the other is subscribed to MAD, so both will choose to launch an annihilatory strike, giving a score of [-10, -10]. Game over.

Now you can talk about decapitating strikes, early warning systems, missile defence systems, etc. all you want, but these don't work for certain. Thus, the only "winning" strategy is to not launch nuclear weapons - which neither side did - meaning they both won (or drew?). You can also talk about the role of the press in shaping the narrative by claiming their side won, for example, the US press running headlines saying things like 'K Bows!',[3] but the fact of the matter is that if you are teetering on the edge of nuclear war, the only "winning" position is one of cooperation to roll-back tensions so we can all keep playing the game.

[1] For the Chicken Game, see: A. Rapoport, & A. Chammah, 'The Game of Chicken', American Behavioural Scientist 10 (1966), pp. 10 - 28.

[2] For MAD, see: J. Gaddis, Strategies of Containment (Oxford, 1982).

[3] Headline of the New York Mirror, see: http://c8.alamy.com/comp/E5GMTM/1962-new-york-mirror-front-page-reporting-the-cuba-missile-crisis-E5GMTM.jpg

EDIT: words