Michael Perez added 17 points and Jerry Evans Jr. scored 15 for the Wolf Pack (14-16, 9-8 Mountain West). Burton began the second overtime with a three-point play and Nevada never trailed again while winning for only the second time in eight games.
Ryan Watkins scored a career-high 27 points and grabbed 14 rebounds for the Broncos (19-10, 9-8). Derrick Marks scored 16 points and Anthony Drmic and Jeff Elorriaga tallied 15 apiece.
Nevada trailed 62-57 with under three minutes left in regulation before Burton's driving hoop and Marqueze Coleman's 3-pointer tied the score with 2:05 remaining. In the first overtime, Nevada led 69-66 after two free throws by Perez with 6.3 seconds before Thomas Bropleh drained a 3-pointer at the buzzer.
Burton, who also had eight assists and six rebounds while playing all 50 minutes, took control in the second session and converted a second three-point play to give the Wolf Pack a 79-75 advantage with 2:09 remaining. He added a driving basket with 1:21 left to again make it a four-point margin.
Watkins got Boise State within 81-79 on a hoop with 54.8 seconds remaining and the Broncos had an opportunity to tie the contest but Coleman swatted away a driving attempt by Marks with 15 seconds left.
Cole Huff drained two free throws with 10 seconds left to make it a two-possession game as Nevada pulled into a tie for fourth place with Boise State in the Mountain West race.
The Wolf Pack led by 14 points 11-plus minutes into the game and appeared to be in control when Burton hammered home a dunk and hit the ensuing free throw to provide a 42-30 lead with 17:04 remaining.
Boise State immediately swiped away the momentum with 10 straight points. Marks scored four points in a span of seven seconds and Drmic and Elorriaga followed with 3-pointers to make it a two-point game.
The Broncos finally caught Nevada at 45 on Elorriaga's fifth 3-pointer with 13:05 left. A short time later, Drmic tied the score with a 3-pointer to start a 9-2 run in which he scored all the points as Boise State took a 54-50 advantage with 8:32 to play.
A basket by Watkins with 3:22 left gave the Broncos a 62-57 advantage but Boise State didn't score again in regulation and trailed for most of the first overtime before Bropleh nailed his clutch 3-pointer.
The Broncos missed 10 consecutive field-goal attempts over a six-minute stretch of the first half and Nevada took advantage and rattled off 10 straight points for a 24-10 lead. The Wolf Pack led 32-24 at halftime.
Watkins' second basket of the contest gave him his 1,000th career point, making him the 25th player in Boise State history to reach the milestone.