The Rainbow Warriors (4-8, 3-4 Mountain West) completed an unlikely comeback after allowing the Rebels (2-10, 1-6) to pull ahead, 35-31, with just 15 seconds to play.
Hawaii began its final drive from the UNLV 42-yard line. Woolsey completed a 22-yard pass to Donnie King on first down and two plays later - with just one second left on the game clock - Woolsey threw a pass toward the front left corner of the end zone and Kemp went up to snag it away from the defender.
UNLV scored the game's first 14 points before allowing Hawaii to reel off 24 consecutive points.
Blake Decker, in relief of starter Jared Lebowitz, entered the game with 12:10 to play and threw three fourth-quarter touchdown passes, the last of which was a 7-yard strike to Taylor Barnhill in the game's final minute. Jonathan Leiva's extra point gave the Rebels a 35-31 lead before Hawaii staged its comeback.
Joey Iosefa carried 35 times and rushed for a career-high 219 yards for Hawaii. He scored on a 9-yard run late in the first quarter and a 1-yard run late in the fourth and also threw a 1-yard touchdown pass to Harold Moleni in the second quarter. Diocemy Saint Juste added 11 carries for 81 yards, including a 52-yard touchdown run in the third quarter.
The Rainbow Warriors rushed for a season-high 348 yards and finished with 578 yards of total offense.
Kemp finished with a game-high six receptions for 114 yards and Woolsey completed 13 of 21 pass attempts for 229 yards with an interception. Ne'Quan Phillips led Hawaii on defense with a game-high 10 tackles, including two sacks.
UNLV threw for 423 of its 474 yards of total offense. Decker threw for 187 yards on 10-of-13 passing and Lebowitz, who was making his first career start, threw for 212 yards on 13-of-30 passing with one interception. Devante Davis led all receivers with 163 yards on five grabs. Peni Vea and Ryan McAleennan had nine tackles apiece to lead the defense.
The lead changed hands five times over the course of the game, including three times in the final two minutes.
The Rainbow Warriors have now won back-to-back games for the first time since 2012, when they beat UNLV and South Alabama to end the season.
The Rebels have now lost their last five games and nine of their previous 10. They conclude the season with a road game against rival Nevada Saturday in Las Vegas.
It was the 24th meeting between the teams dating back to 1969. Hawaii has won 10 of 12 games in Honolulu, including the past four and now leads the all-time series, 15-9.