Victories haven't come easy for the Jazz during this rebuilding season.
"We realize we're not there yet individually, where we can just play for ourselves and still win. We need to play as a team," Hayward said.
The Jazz had 22 assists on their 33 baskets and several other unselfish passes led to 35 attempts from the free throw line.
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.
"Alec attacks the rim and he's got to be one of the best in the league at driving to the basket," Hayward said. "He finishes so well that he scores or gets fouls almost every time."
Two exceptions occurred in the waning moments -- swarming Pelicans forced Burks to travel and then 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.
"That's exactly what we wanted. We had Tyreke (Evans) with the ball looking to find a shooter and he found Darius. It was the best look we could ask for, and it just didn't go in," Brian Roberts said.
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. Burks made all 13 of his free throw attempts.
Derrick Favors had 20 points and nine rebounds for the Jazz. He thrived without Davis playing at full strength.
"There was more room in the post," Favors said. "We had people move around instead of standing still, so it was a big help."
Roberts had 18 points while Davis finished with eight points and six rebounds in 23 minutes as the Pelicans dropped their fourth straight following a five-game win streak.
The game was tight throughout, but the Jazz extended their lead to double digits near the end of the third when Hayward hit a 3 and followed with another jumper to make it 73-63.
But the Pelicans responded despite playing an unusual lineup that was sitting at the end of the bench earlier this season before the team was decimated with injuries. Eric Gordon, Ryan Anderson, Jrue Holiday and Jason Smith -- a group that averages nearly 60 points a game combined -- all are sidelined.
New Orleans turned to Morrow, who trimmed the Jazz advantage to 82-79 on a jumper in the lane with 7:16 to play. Morrow also had a dozen points in the fourth quarter on 5-for-5 shooting to give the Pelicans a chance. The guard has scored at least 17 points in five of his past six games.
The Jazz again built the lead to 10 points when Hayward made a pair of free throws. The Pelicans then made their last-gasp comeback that fell just short.
On one play in the third quarter, Davis blocked Favors' shot and then saved the ball from going out of bounds. But as the two big men pursued the ball, Davis was sandwiched between Favors and the fans courtside. Davis gingerly ran back on offense and soon after left the court to be evaluated.
The training staff confirmed Davis had back spasms but he returned to play in the fourth quarter, but wasn't on the court when the Pelicans made their final push.
"He wanted to play and I just said no. We're not in the playoffs. Every game for me is a championship game, but I've got to be smart," Pelicans coach Monty Williams said.
Davis played only 11 minutes against the Nuggets and his absence was felt as the Pelicans allowed season highs in points (137) and field goal shooting (56.5).
After three days' rest at home, the Jazz also took advantage with 51.6 percent shooting.
"We talked about being able to finish games and we still have the energy to compete. It's been a long year for these guys and we've been through a lot," Utah coach Ty Corbin said.
Stiemsma got a technical foul for arguing with the officials midway through the third period. ... The Pelicans have lost 12 of 15 on the road. ... After a long slump, Hayward is averaging 21 points a game in his past six.
The Hawks get their first postseason win in Boston since 1988, finishing off the Celtics 104-92, for a 4-2 series victory.
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The Hawks scored 39 points in the third quarter to pull away from the Celtics and won Thursday night's Game 6 104-92 to advance to the second round, where they'll face the Cleveland Cavaliers.