Label a position a winning position if there is at least one move to a losing position, a losing position as one with no moves to non-winning positions, and a drawing position as one where there is no move to a losing solution, but there is a move to a drawing position. The final position(s) are unknown, but they are not . The first player has a winning strategy on a position when it is a winning position, and the second player has a winning strategy if it is a losing position (this should at least be obvious). Therefore, the only time when neither player has a winning strategy is in a drawing position.
Consider such a drawing position. By definition there is a move to another drawing position. However, that creates an infinite sequence of moves, which contradicts the original assumption that the game is finite. Therefore there are no drawing positions, and every finite game that cannot end in a draw has a winning strategy for one of the players.
Note: This argument doesn't work for infinite games. There are some (highly interesting) infinite games that cannot end in a draw, but neither player has a winning strategy.