Yes, to be a stickler, it should be "power of you and me," since that phrase is the object of the subject and verb "I believe." And "me" is the objective case, while "I" is the subjective case. (Similarly, you wouldn't say, "Me believe in the power of I.")

It's a common mistake (often in songs for the rhyme), the same as when people say "between you and I" when it should be "between you and me." (That has to do with the objective case after the preposition "between.") They think it sounds more proper. My theory is it comes from years of parents and teachers correcting us when we say, "Johnny and me are going outside to play." "No, Billy: Johnny and I are going outside to play."

So when it comes to saying, as we should, "Give the ball to Johnny and me," some little voice in our head tells us it must be "Johnny and I."

The easiest way to test it is to take the other person out of the equation: I am going outside. Give the ball to me. I believe in (the power of) me.

But of course the song has to rhyme with "together we'll fly." Otherwise it would have to be, "I believe together we'll flee / I believe in the power of you and me." Which somehow doesn't reflect the Olympic spirit.