Tyler Cavanaugh added 12 points for the Demon Deacons (15-13, 5-10 Atlantic Coast Conference).
They erased an early 10-point deficit, shot 55 percent against the league's second-stingiest defense and held on to avoid just their third winless February since the school started keeping complete records in 1951.
Damarcus Harrison scored 15 points for the Tigers (17-10, 8-7). They pulled within two points on Jaron Blossomgame's three-point play with 29.1 seconds left.
After Cavanaugh made it a four-point game with two free throws with 20.7 seconds left, Rod Hall missed on a contested drive.
Travis McKie added a free throw with 8.2 seconds remaining.
K.J. McDaniels - the league's third-leading scorer - had 10 points on 3-of-11 shooting. The Tigers' NCAA tournament hopes took a serious hit.
Harrison hit back-to-back 3-pointers to pull Clemson to 56-54 with 2 minutes left.
But with his team down by two with 1:39 left, McDaniels missed two free throws. The rebound went to Codi Miller-McIntyre, and the Wake Forest guard jetted toward the bucket and hit a layup through contact from Hall. His free throw made it 59-54 with 1:34 left.
Hall finished with 10 points for Clemson, which managed just one field goal during the 6 1/2-minute span in which Wake Forest took the lead for good and fell to 2-19 at Joel Coliseum.
Arnaud William Adala Moto scored 11 points. Miller-McIntyre finished with 10, three days after the guard was held scoreless in a 33-point loss at rival North Carolina.
The Demon Deacons outscored Clemson 13-2 down the stretch and went up 55-46 on Thomas' layup with just under 4 minutes left.
That came after they erased a 10-point deficit and took a 35-34 lead with 15 1/2 minutes left on Adala Moto's alley-oop lob to McKie, and it stayed tight until the Demon Deacons' big run.
Hardly anybody saw this result coming, after Wake Forest spent the entire month of February in a free-fall with four of six losses in the month coming by double figures.
Meanwhile, Clemson had won two straight and arrived with a shot at the NCAA tournament and had plenty to play for.
And early on, the Tigers appeared on their way to their sixth straight victory in the series after they held Wake Forest without a rebound for a stretch of 8 1/2 minutes and kept the Demon Deacons completely off the offensive glass in the first half.
Yet the Tigers' only double-figure lead came with their best player on the bench. McDaniels sat for most of the final 5 minutes of the half with two fouls. Clemson went up 30-20 on Harrison's short jumper with about 1:45 left before the break.
Follow Joedy McCreary on Twitter at https://twitter.com/joedyap