Burks had 12 of his points in the fourth quarter and Hayward added eight final-period points. The Jazz needed all of it to overcome the reeling Pelicans, who played without Anthony Davis down the stretch due to back spasms.
After two exceptional defense plays when the Pelicans forced a turnover and Greg Stiemsma blocked Burks' layup, Anthony Morrow, who had a game-high 26 points, hit a 3-pointer to draw New Orleans within 95-93.
Hayward made one of two free throws and the Pelicans drew a play that produced a wide-open shot for Darius Miller that rimmed out.
Brian Roberts made another 3 with 1.7 seconds remaining, but Hayward and Burks each made two more free throws to clinch the victory for the Jazz.