It was Toronto's first win in the series since December 22, 2004.
Bargnani, who played his second game after missing the previous six with a strained calf, scored all of his points in regulation. He played 3:55 in the first overtime before sitting the rest of the game.
Jose Calderon knocked down a three-pointer with 13.3 seconds remaining in regulation for Toronto, tying the game, while Paul Millsap buried a three with 3.0 seconds left in the first overtime to tie the game for Utah.
The Jazz played without leading scorer Al Jefferson because of an inflamed right ankle, but still had an 18-point lead in the first quarter.
"It's a tough loss for us because we know we should have won the game," said Millsap. "(It fell apart) in the second quarter. Our defense was poor."
It was the second time in two days that the Raptors snapped a long losing streak to another team. They beat Phoenix, 99-96, in Bargnani's return on Tuesday for their first win over the Suns since February 10, 2004.