Two kids are playing a game. Right after the game is over one of them shouts happily:
"I won! I won!"
or should he better say "I win! I win!"
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As a celebration of winning a game just played, the use of I win is grammatical and probably "more appropriate."
It is common in this context (of right after the game is over) for native speakers to say I win or we win or you win; in this case the act of winning is seen as being in the present.
But since the game play is actually over, I won is also grammatically correct.
The window of time does not last long to say I win in respect to a game just played. Like you said, it has to be right after the game. A person can be in the act of winning only so long. When you pick up the game pieces to start a new round, that moment is pretty much past.
Another use of the present is I win every time we play this game. In this case the present simple describes an ongoing habit. So a person can say this before, during or after a game.