L.J. Rose and TaShawn Thomas donated 16 and 14 points, respectively, in the triumph, with Thomas also contributing nine rebounds, four assists and four blocked shots. Danuel House scored 11 in Houston's (17-15) fifth win in its last six games.
The Cougars await the winner of the Louisville vs. Rutgers game.
Cannen Cunningham had 14 points in a reserve role for SMU (23-9), which has lost three in a row for the first time all season. Markus Kennedy finished with 13 points and nine rebounds, while Nick Russell added 10 points. Nic Moore shot 0-for-9 from the floor but handed out a game-high eight assists in the setback.
A pair of turnovers on Houston's first two possessions led to two easy SMU buckets and an early 4-0 edge. The spread reached 7-0 on a Sterling Brown corner triple and Houston took a timeout at 17:37 to regroup.
Stiggers injected some life into Houston's offense out of the break, hitting three 3-pointers to pull the Cougars within 14-13 midway through the opening period. Rose's triple at 8:12 gave Houston its first lead.
Russell netted eight points during a 12-4 SMU flurry to close the half and the Mustangs carried a 38-29 margin into the locker rooms.
Brown's layup just over two minutes into the second made it 42-32 before Thomas sparked a 16-4 Houston push. Thomas scored seven during the surge, while House and Stiggers hit consecutive 3-balls on the back end for a 51-46 Cougars edge at 12:29.
The lead held the rest of the way, with Houston sinking a number of timely baskets down the stretch. Thomas' layup with 1:23 remaining was the big one, as it gave the underdog Cougars a 64-59 advantage.
It was a three-point lead after two Moore free throws. House missed the front end of a 1-and-1, but Thomas swatted Kennedy at the rim with 17 seconds left. House then sealed it with two from the stripe.