The Warriors (6-6) racked up 482 offensive yards, including 358 passing, to snap a three-game losing skid.
Graves threw touchdown passes of 17 and 12 yards -- both to Trevor Davis -- and ran in another score from 15 yards out in his first career start. He left the game late in the third quarter with a head injury.
Hawaii held a 21-14 halftime lead and extended it on Graves' second scoring connection with Davis early in the third quarter. Tulane got a safety when it tackled Iosefa in the end zone later that quarter. A touchdown run by Orleans Darkwa in the fourth quarter made it 28-23.
The Green Wave (2-11) got no closer. Iosefa sealed it with his second touchdown run midway through the fourth quarter.
The Warriors -- an 18-point favorite by most Las Vegas betting lines -- were playing in their first game since news broke of an alleged point-shaving scandal involving their football players.
An anonymous letter making the claims was sent to the school Nov. 3, accusing unnamed players of intentionally playing poorly to affect the score as part of a gambling scheme.
Hawaii scored on the game's opening drive with a 17-yard pass from Graves to Davis.
Tulane tied it on an 11-yard touchdown run by Albert Williams later in the first quarter, but Graves answered for the Warriors with a 15-yard scamper that made it 14-7.
The Green Wave used a 13-play drive that covered 68 yards and was capped by a 1-yard plunge into the end zone by Darkwa to tie it at 14 early in the second quarter.
Hawaii returned the favor on its ensuing possession and reclaimed the lead on Iosefa's 1-yard run with seven minutes left until halftime.
Tulane failed to convert on two first-half field goals by Cairo Santos. A 39-yard attempt sailed wide left and a 40-yarder on the final play of the second quarter was blocked by Hawaii's John Hardy-Tuliau.
Graves finished 21-of-28 passing for 260 yards with two interceptions. Shane Austin, who replaced him late in the game, was 9 of 10 for 98 yards. Corey Paredes led all players with 13 tackles for Hawaii.
It was the first meeting between the teams since 2002, when Tulane beat Hawaii 36-28 in the Hawaii Bowl.
Hawaii can become bowl-eligible if it beats BYU next week. If the Warriors win, they would host a Conference USA team in the Hawaii Bowl at Aloha Stadium on Christmas Eve.