Ebon Harris made a 3-pointer with 31 seconds left in regulation to rescue West Virginia after the Mountaineers had let a 14-point lead get away in the second half.
Harris led the Mountaineers with 28 points and he scored six of them in the extra period.
Oklahoma climbed within 87-86 on a layup by Jordan Woodward with 1:40 to go, but the Sooners could not score again. Staten boosted the West Virginia lead to three points with 1:03 to go and then made 1-of-2 from the foul line with 29 seconds left.
West Virginia has won five out of six to reach 14-9 for the season. The Mountaineers are 6-4 in the Big 12 Conference, putting them in a four-way tie for third place with Oklahoma, Iowa State and Kansas State.
Staten added 20 points and 10 rebounds while Terry Henderson had 17 for West Virginia.
Woodward paced the Sooners (17-6, 6-4) with 23 points.