Payten Jason scored on a 2-yard run and Cairo Santos added a 39-yard field goal as the Green Wave (2-2) snapped a six-game road losing streak dating to last season. Alex Wacha iced the game with an interception in the final minute.
Rutgers starting quarterback Tom Savage, who came into the game with bruised ribs, was hurt late in the second quarter and did not play in the second half. The school did not disclose his injury.
The loss was the second straight for the Scarlet Knights (2-2), and this one came before a disappointed homecoming crowd who saw a struggling offense and penalties lead to another setback.
Rutgers had an early 95-yard kickoff return for a touchdown by Joe Lefeged nullified by a holding penalty, and another one against halfback Kordell Young on the final drive cost them a first down at the Green Wave 28.
Limited to 245 total yards, Tulane used a little trickery and took advantage of two short punts by Teddy Dellaganna to score its touchdowns on short drives.
The big play came early in the third quarter when Kemp, a receiver who started six games at quarterback last season, lined up in a wildcat position. He took the snap and threw a lateral to Banks standing near the Tulane sideline. Kemp and the line went left and Banks tossed the ball back across the field for an easy touchdown and a 17-7 lead.
Dodd, who finished 13-of-29 for 176 yards, got Rutgers back in the game with his third-down touchdown pass to Sanu, who stepped out of a tackle at the 5-yard line.
But Tulane held the rest of the way and scored the upset.
After doing little on offense in its opening three games, Rutgers appeared ready to roll when Sanu took a snap in the wildcat and went untouched for his touchdown on the second play from scrimmage.
The Scarlet Knights' old record of 90 yards was held jointly by Ray Rice (2008) and Chad Bosch (1996).
The rest of the half, Tulane held Rutgers at bay, thanks in large part to some costly penalties.
After Santos cut the lead to 7-3 with a 39-yard field goal early in the second quarter, Lefeged ran back the ensuing kickoff, only to see a flag on the ground.
Not only were the Scarlet Knights forced to start the drive at their 18, but they went three-and-out, and Dellaganna shanked his punt 20 yards.
Tulane took over at the Scarlet Knights 33 and needed only five plays to take the lead, with Jason spinning the final two yards for the touchdown.
The Green Wave's final TD drive started at the Rutgers 43 after a 21-yard punt by Dellaganna.