Burke's runner at the regulation buzzer bounced off the back of the iron.
Burke scored five points to spark a 16-0 Jazz run in the fourth quarter that erased Sacramento's nine-point lead. Hayward capped the run with back-to-back jumpers and a free throw to make it 91-84 with 2:59 to play.
"It hurts. We had the game," Hayward said.
The Jazz and the Kings entered the night tied with Milwaukee for the fewest wins (4) in the NBA. Four of the defeats in the Kings six-game skid were by a combined 11 points and they looked ready to turn a corner. McLemore followed his career-high 20 points in Los Angeles with 15 points and nine rebounds against the Jazz.
All the losing precipitated an impromptu team meeting at a Salt Lake City hotel Saturday morning. Sacramento coach Mike Malone asked his players for input on how they could change course.
"I said we need to be tougher to guard," Thomas said. "Everybody spoke on it and Coach accepted it. And it worked. We ran a couple different plays to get DeMarcus the ball and it opened the floor."
Cousins overcame early foul trouble and was extremely efficient in 28 minutes, going 10 of 15 from the field. Favors, a potential foil for Cousins who had started all 21 games this season for the Jazz, sat out with a sore lower back.
"We got their bigs in foul trouble. We caught a little bit of a break with Favors not being healthy, so we wanted to establish DeMarcus in the post, feed off of him," Malone said.
NOTES: Cousins committed two fouls in the first minute of the game and Utah's Mike Harris had five fouls before halftime. Harris and Jeremy Evans both fouled out in regulation and Enes Kanter fouled out in overtime. ... In the first quarter, the Kings twice intentionally fouled Andris Biedrins, a career 50 percent free throw shooter. Biedrins made 1 of 6 from the line. ... Cousins lost his trademark headband late in the fourth quarter on a hard foul by Kanter.