First I'm going to mention another game. Noughts and crosses or tic-tac-toe. Everyone has played it and everyone older than 8 or so figured out how to play optimally. I'm going to claim that the nine card game is tic-tac-toe in a very precise sense.
Consider the magic square.
4 9 2
3 5 7
8 1 6
It's known for every row, column and diagonal summing to 15. A little work will convince you that these are also the only ways to sum to 15, with exactly 3 distinct integers between 1 and 9.
As we all know how to play tic-tac-toe, we're done.
(edit: the middle row of the magic square appeared off-center,think it should be fixed now)