Burke scored just three points on Monday in a loss to Miami, the lowest total of his rookie season, but he came back with a monster effort against the Magic to push the Jazz to a rare victory.
In addition to his 30 points, Burke grabbed seven rebounds and had eight assists.
Orlando moved within two points when Glen Davis made a pair of free throws with 1:01 to play and the only other scoring came when Hayward's two foul shots with 2 seconds remaining secured the win.
Hayward added 15 points, while Derrick Favors chipped in 14 and 11 rebounds. It was the seventh win for the Jazz in 28 games this season.
Jameer Nelson scored 17 points to lead Orlando and Nikola Vucevic had 14 points and 14 rebounds.