Yavi jumped for joy after crossing the line, having overtaken Chemutai with an explosive final sprint that left the Ugandan who had led most of the race unable to respond.
Chemutai was in tears after Yavi, 24, stole the finish to add the Olympic title to last year's world championship, setting an Olympic record of 8 minutes 52.76 seconds.
Kenya's Faith Cherotich, 20, claimed bronze on her Olympic debut.
Kenyan world record holder Beatrice Chepkoech was shoulder-to-shoulder with Chemutai but faded in the last lap and ended sixth, while France's Alice Finot was delighted with a fourth-place finish in a very fast race.
Chepkoech holds the world record of 8:44.32 but has never won an Olympic medal in the event.