Elpheage ran back Hawaii's next punt 56 yards, giving Tulane the ball at the Hawaii 17. That set up the go-ahead touchdown, a 1-yard run by J.P. Losman. Elpheage was voted the game's outstanding player after returning four punts for 143 yards.
The Green Wave (8-5) came in as a double-digit underdog but threatened to turn the game into a rout when Mewelde Moore scooted 25 yards late in the third period to make it 26-14.
However, Hawaii (10-4) came back behind Shawn Withy-Allen, who filled in for an injured Timmy Chang. Withy-Allen hooked up on a 57-yard scoring pass with Justin Colbert to make it 26-21.
But Losman scored his second rushing TD, a 3-yarder, to extend the lead to 34-21. Withy-Allen and Colbert combined on a 31-yard scoring play to narrow the gap with 5:12 left, but Tulane put the game out of reach when Roxie Shelvin sacked Withy-Allen in the end zone for a safety to make it 36-28 with 3:02 left.
Losman and Moore carried the Tulane offense. Losman completed 20 of 39 passes for 240 yards. Moore led the team in rushing with 30 carries for 116 yards, and receiving with 6 catches for 80 yards. At the half, Moore had only 22 yards rushing.
Tulane, which had not played since Nov. 23, appeared sluggish in the first half and mustered only 172 yards in total offense. But Elpheage got the Green Wave started with his returns.
Hawaii jumped out to a 14-0 lead in the first half behind short scoring runs by Thero Mitchell and Josh Galeai, but Tulane scored 26 unanswered points - including 20 in the third quarter.
A pair of field goals by Seth Marler got Tulane going, closing Hawaii's lead to 14-6.
Hawaii, the nation's top passing offense, was without Chang for most of the game after he injured the thumb on his throwing hand in the second quarter. Chang completed 14 of 21 passes for 124 yards.
Withy-Allen finished with 239 yards. Colbert was Hawaii's top receiver with nine catches for 158 yards.