With the Jazz trailing 88-86, Gordon Hayward dribbled through the center of the Magic defense and found Burke in the corner. The big shot snapped Utah's six-game losing streak and extended Orlando's skid to eight games.
Down the stretch, the game turned into a battle between Burke and Victor Oladipo, two rookies drafted in the top 10 of last summer's draft.
Oladipo, the second selection, scored 19 points for the Magic. He made a jumper and then added two free throws with 21 seconds to play to give Orlando an 85-82 lead.
Hayward, who had 14 points, made two free throws and then assisted Derrick Favors and Burke for the game-winner, all in the final 20 seconds.
Richard Jefferson scored 17 of his 21 points in the first quarter and Favors, Enes Kanter and Diante Garrett all scored 11 for the Jazz.
Aaron Afflalo scored 19 points for the Magic. He made three free throws in the final 20 seconds but his miss with 12 seconds left opened the door for Burke.
Favors knocked the inbounds pass away after Burke's shot, leaving just 0.4 seconds on the clock. The Magic inbounded to Afflalo, who missed a desperation 26-footer that may have left his hands after the buzzer sounded anyway.
At times, it seemed neither team wanted to win. There was no exceptional defense on display but both squads struggled to hit open shots, neither topping 40 percent from the field.
Burke, who was taken ninth and had some battles with Oladipo in college, saved his best for last. He was embarrassed on a couple of Oladipo's drives to the hoop earlier in the game and made only five of his 15 shots.
Just as he did at Michigan as the consensus national player of the year, Burke showed he's not afraid to take the big shot with two 3-pointers in the final three minutes.
Oladipo, meanwhile, also struggled with his shot, going 5 for 14. But had a far better performance than his three points while shooting a season-worst 1 of 12 in the 86-82 loss at Orlando in this season's first matchup on Dec. 18.
The Magic are 4-23 against Western Conference teams, including 0-14 on the road.
Jefferson made four 3-pointers and had 17 points in the first quarter to spark the Jazz to lead by as many as 14 points in the first half before the Magic cut it to 49-43 at halftime.
The Jazz have won seven in a row against the Magic, one of the few teams they can consistently beat these days. Orlando came along just at the right time for the Jazz, who had dropped 11 of 12, including six straight by an average of 14.0 points.
The teams' ineptitude was on full display even as the game was up for grabs. During a five-minute stretch spanning the third and fourth quarters, the teams combined for 12 empty offensive possessions.
Jameer Nelson rested his sore left knee for the second straight game, giving Oladipo more opportunity to play. But the Magic were absent one more playmaker when they needed to make just one more play.