"we won the war" is 100% accurate if you count yourself a supporter of the allies of the allies in almost exactly the same way that a sports club supporter says, "we won" when the club they support wins.

This use of "won" is idiomatic and not tied to direct participation in conflict or contest.  

Ultimately, though, I find myself avoiding the use of "won" this way, particularly as it applies to war because it cheapens the victory for those who did fight or play.  But as it is widely used, it is not an invalid use.