The Mountaineers had grabbed their second lead of the fourth quarter when Geno Smith found Stedman Bailey on a 40-yard scoring strike, but the 2-point conversion failed, leaving West Virginia with a 49-44 lead.
Jones then led the Sooners (8-2, 6-1 Big 12) on a six-play, 54-yard drive, capped by the scoring strike to Stills in the final minute -- his sixth TD toss of the game -- handing Oklahoma its third straight win.
Jones finished with 554 passing yards on 38-of-51 throws. Stills made 10 grabs in all for 91 yards.
Jalen Saunders led the Sooners' receivers with seven catches for 123 yards and a score and Damien Williams picked up 163 yards from scrimmage with a TD.
Tavon Austin ran for 344 yards and two scores for the Mountaineers. Smith completed 20-of-35 passes for 320 yards, two scores and two interceptions for West Virginia (5-5, 2-5), which lost its fourth straight.
2014: The Year in Music [PHOTOS]