Marvin Williams and Hayward scored 11 points each to lead the Jazz, who lost starting point guard Mo Williams with a sprained right ankle.
The Kings, 0-4 on the road this season, raced to a 23-13 lead in the first eight minutes.
Brooks scored six points during a 12-4 run that pushed Sacramento in front, before Utah rallied behind the play of Marvin Williams and Hayward.
Williams finished the first quarter with nine points, while Hayward scored 11 during a seven-minute stretch as the Jazz fought back into the game.
They trailed, 47-42, after the Kings' Jimmer Fredette hit a 3-pointer and Jason Thomspon scored on a tip-in.
Utah tied the game, however, with a layup by Marvin Williams and a transition 3-pointer from Randy Foye with 3:23 left.
Another 3-pointer by Foye gave Utah a 50-49 lead with less than a minute left, but Evans scored on the Kings' final two possessions to give them a one-point halftime lead.
The Jazz led 62-61 before going cold midway through the third quarter. Evans and Brooks hit 3-pointers during a 10-0 run that gave Sacramento a nine-point lead.
It was 80-73 heading into the wild fourth quarter, when the Jazz staged their improbable comeback.