Demarco Nelson, who missed all of last season due to academic reasons, intercepted Tanner Lee's pass two plays later to seal the victory.
With Tulsa down 28-20 with 2:53 remaining in regulation, Evans found Keevan Lucas for a 3-yard score and hit Joshua Atkinson for the tying 2-point conversion.
On Tulane's ensuing drive, Sherman Badie's 73-yard run set up the Green Wave with a first-and-goal from the 8 but, three plays later, Andrew DiRocco's 21-yard field goal attempt missed wide left.
The teams exchanged field goals in the first overtime.
In the season opener and the American Athletic Conference debut for both teams, Devon Walker returned to the site where, as a team captain and senior safety for Tulane, he suffered a cervical spine fracture in 2012.