The power Harry had was love.
It wasn't all about killing Voldemort - it was about stopping him and his followers. Harry's sacrifice - walking wilfully into the Forbidden Forest to protect the defenders of Hogwarts - kept them safe them from Voldemort when he returned to the castle. Harry said this himself:
I've done what my mother did. They're protected from you. Haven't you noticed how none of the spells you put on them are binding? You can't torture them. You can't touch them.
The result of this was Voldemort, alone, his Death Eaters fled, killed or captured. At this point, Voldemort had lost. He had nothing left - no Horcruxes, nobody to fall back on. And the entirety of the Order of the Phoenix, countless other witches and wizards, and an army of centaurs was ready to tear out his throat.
The happy accident that gave Harry command of the Elder Wand was icing on the cake, and gave us the poetic justice of Voldemort killing himself, but that was irrelevant. He had already lost. The prophecy stated that Harry's power over Voldemort would allow him to defeat the Dark Lord, and that's exactly what happened.