With the Cougars going cold down the stretch, UH nearly coughed up a double-digit halftime lead to San Diego State, but Houston prevailed, 67-65.
Gray's go-ahead layup with just one second left on the game clock sealed it for Houston.
Gray scored a game-high 39 points, which is the most points scored in a debut NCAA Tournament game.
The Cougars will take on the Michigan Wolverines on Saturday. Tip-off time is set for 8:40 p.m.
VIDEO: UH Cougars excited heading into locker room after big win vs. SDSU
UH Cougars excited after big win vs. SDSU
VIDEO: Coach Kelvin Sampson salutes the UH faithful after win
UH coach Kelvin Sampson walks to locker room after big win
VIDEO: UH star Rob Gray sprints to locker room after 39-point performance
UH's Rob Gray sprints into locker room after big night