If the game was a group activity, like a team sport or something you played as part of a team with one or more teammates ("after WE won the game"), it would be common to say: - "I'm really happy (that) we won."

They are all correct except for 'I am really happy to win'. This is because the infinitive 'to win' places the action 'win' after 'I am happy' and, of course, it should be before it. You could say 'I was really happy to win'.

They are all grammatically correct; however, ‘I’m really happy to win’ sounds out of context because we’re talking after the game, so ‘I’m really happy to have won’ sounds more natural.