Payne scored nine points in the second half. Wallace, who was averaging 3.6 points, matched his season high — all in the second half.
Auburn mounted the comeback even with leading scorer Frankie Sullivan missing his first 11 shots in a 1-for-13, four-point performance.
"We finally didn't let our offensive struggles affect our defense," said Tigers coach Tony Barbee, who had lost his first four games against Alabama.
Trevor Releford and Rodney Cooper led Alabama with 11 points each.
Trevor Lacey fouled out with six points, half his season average, and Levi Randolph missed all seven of his shots.
The Tide made just 6 of 25 shots (24 percent) in the second half and was 2 of 19 from 3-point range for the game.
Auburn was even colder to start before hitting 12 of 21 shots (57.1 percent) in the second half.
"I didn't have to light a fire at all," Barbee said. "I thought we were playing hard."
The Tigers, who had just three baskets by halftime, came out of the locker room with a 25-5 surge.
The Tide had a scoring drought of 9:21 after Cooper's early 3-pointer
"They outhustled us for loose balls," Grant said. "They got out in transition and got layups. Their effort kept going up and we didn't match that. We can't do that.
"We need to look in the mirror because this was completely on us in terms of the energy we're supposed to play with in a game that means as much as this one meant."
Chris Denson gave Auburn a 38-28 lead on a drive with 6:25 left. The Tigers then scored five straight points after Cooper broke the scoreless stretch with a jumper.
Wallace scored on a drive and two free throws to push the lead to 44-31 with 2:10 remaining.