The catches by Hicks, his first of the game, were for 38, 27 and 2 yards, the last one for the touchdown to put the Red Raiders (7-4, 5-3 Big 12) up 14-9.
Hicks caught his second TD pass, a 14-yarder, midway through the fourth quarter to make it 24-15. Taurean Henderson sealed the win on a 6-yard run with 2:52 remaining in the game.
The Cowboys have dropped four of its last six games following a 5-0 start.
Cumbie, who came in averaging 394 yards passing, was 31-for-46 with an interception. His previous low of 294 yards came two weeks ago in a 42-35 overtime loss at Texas A&M.
OSU (7-4, 4-4) took a 15-14 lead when safety Jaime Thompson intercepted a pass from Cumbie and returned it 68 yards for a touchdown with 3:06 remaining in the third quarter. The Cowboys attempted a 2-point conversion but failed after Cowboys quarterback Donovan Woods fumbled the snap.
Tech regained the lead, 17-15, after a 13-play drive stalled at the 5 and Alex Trlica hit a 22-yard field goal with 9:43 remaining in the game.
Cumbie had a season-low 127 passing yards on 16-of-22 in the first half. His previous first-half low was 132 in a 42-17 win over Baylor earlier this month.
The Cowboys threw just one pass in the first half. With about seven minutes remaining in the second quarter, Woods completed a 5-yarder to Vernand Morency on third-and-8, forcing OSU to punt in its only three-and-out of the half.
Woods finished 6-for-9 for 60 yards.
OSU led 9-7 at halftime. The Cowboys ran for 122 yards in the first half, 86 of those from Morency. Morency finished with 108 yards on 23 carries, well below his 149-yard average.
Tech's defense held the Cowboys scoreless in the fourth quarter and to 176 yards rushing for the game. They ranked eighth nationally coming into the game, averaging 251.9 yards per game.