Q: If, for my one play, I play something that makes me win with the Cute Fuzzy Alien Creature, does it move to the next person before I can win?

A: Unlike some cards which declare that they do something “When the Goal changes” which would be simultaneous with the Goal play, simply taking your last play does not instantaneously end your turn. So there’s plenty of time for the Goal to take effect, and for you to win. In effect, you win ON your turn, so your turn doesn’t “end”… the game itself ends, with you as the winner.