West Virginia (11-11, 4-5 Big 12 Conference) swept the season series and sent the Longhorns (10-12, 2-7) to their seventh loss in nine games. West Virginia has won two straight for the first time since late December.
Texas trailed by as many as 11 early in the second half but went ahead 49-47 on two free throws by Felix with 8:21 left.
But the Longhorns were held to nine points -- and two field goals -- the rest of the game.
West Virginia scored five straight points, including Eron Harris' free throw that put the Mountaineers ahead 59-54 with 25 seconds left.
West Virginia had three turnovers in the final 2 minutes, only to see the Longhorns give the ball right back with turnovers, missed shots and fouls.
Still, Sheldon McClellan gave Texas some chances at the very end. He made two free throws after being fouled by Kilicli on a 3-point try with 11 seconds left to draw Texas within 59-56.
After Aaric Murray made a free throw for West Virginia with 9 seconds to go, McClellan was fouled on another 3-point attempt, this time by Murray with under a second remaining. McClellan made the first two free throws and Texas called timeout. McClellan missed the final one on purpose and Murray got the rebound as time expired.
Texas held a 36-29 rebounding advantage but was held to 35 percent shooting (21 of 60) and was outscored 20-11 from the free-throw line.
Harris finished with 13 points for West Virginia.
It was the type of scrappy game that fit how both teams are clawing for any win at this point during disappointing seasons.
Texas has made the NCAA tournament each of the last 14 seasons under Rick Barnes, while West Virginia has gone to five straight in each of Bob Huggins' five seasons.
Ted Kapita was the top-rated recruit from West Virginia in the class of 2015 and recently played at DME Academy in Florida.
With the Suns drafting Tyler Ulis, they now have five guards on their roster who attended Kentucky.
NBA fans will be looking forward to their newest players making plays like this next season.