Doege shook off windy conditions en route to the big performance, in which he went 32-of-42 through the air with an interception for the Red Raiders (5-1, 2-1 Big 12).
Three of his scoring strikes went to Darrin Moore, who hauled in nine passes for 92 yards.
Jace Amaro made five grabs for 156 yards and a score for the victors.
West Virginia's Geno Smith, meanwhile, managed to complete only 29-of-55 passes for 275 yards as the Mountaineers (5-1, 2-1) saw an eight-game winning streak snapped.
Stedman Bailey made six catches for 56 yards and a touchdown but sat out the second half with an ankle injury.