The game heated up after Arron Afflalo was ejected and the Jazz stormed back from a 22-point deficit to tie the game in the final minutes.
Lawson, who also had 12 assists, made the game-winner from the left wing between two Jazz defenders.
Hayward scored 25 points, Burks had 22 and Burke chipped in 18, but Utah's losing streak hit six games " the longest skid since March 10-19 last season.
At Washington, John Wall had 18 points and 13 assists as the Wizards made their first seven 3-pointers to beat the Miami Heat 107-86.
The Wizard held the Heat to 2 for 22 from beyond the arc for the game.
Rasual Butler scored 23, and Marcin Gortat added 15 points and 10 rebounds for the Wizards, who never trailed and led by 21 in the first half and 25 in the second.
Chris Bosh led Miami with 21 points, and Dwayne Wade had 20 in his second game back from an injured right hamstring.
The Los Angeles Clippers downed the Minnesota Timberwolves 127-101 in the most successful road trip in club history after Blake Griffin and J.J. Redick each scored 23 points before sitting out the fourth quarter.
DeAndre Jordan had 13 rebounds and 11 points for the Clippers, who won their fifth in a row following their 6-1 trip and beat Minnesota for the 10th straight time.
Shabazz Muhammad had 18 points and 10 rebounds for the Timberwolves.