Can you prove who wins a game without knowing how?

Yes, for some games, with a clever trick called strategy-stealing. Take a game where an extra move can never hurt you. Suppose the second player had a guaranteed winning strategy. The first player could make an arbitrary first move, which is no disadvantage in such a game, and then play that same strategy. Both players would be guaranteed to win, which is absurd, so the assumed strategy cannot exist. The first player can therefore win (or possibly draw) without anyone constructing a strategy. That is the catch: the proof says a winning strategy exists and gives no information about what it is.

The classic case is Hex, a two-player game in which players try to connect opposite sides of a rhombus board of hexagonal cells. Draws are impossible in Hex because of the topology of the board, so one side must win, and John Nash's argument shows that on symmetrical boards it is the first player. His fellow players called the game Nash or John, because it could be played on hexagonal bathroom tiles. Nash invented strategy-stealing to prove this, but did not publish the method.

This gives game theorists a ladder of ways to "solve" a game. An ultra-weak solution proves who wins under perfect play, without details. A weak solution gives each player an algorithm that achieves at least the optimal outcome from the start. A strong solution finds the best play from all legal positions. Many game theorists consider ultra-weak proofs the deepest, while strong ones often proceed by brute force. Tic-tac-toe is a simple strong solution: a draw with perfect play.