It allowed the visiting Crimson and Cream clad to escape after surrendering a ridiculous 778 yards of total offense, including 344 on the ground and 572 all-purpose yards to Tavon Austin.
And OU even came out on top despite squandering a double-digit lead in the second half.
The Sooners led 38-24 after Jones' second of four touchdown passes to Stills at the 11:54 mark of the third quarter, but West Virginia rattled off 19 unanswered points, including a 4-yard touchdown by Austin and a pair of touchdown passes from Geno Smith to star wide out Stedman Bailey, from four and eight yards, respectively.
That made it 43-38 with 7:12 remaining before the teams rode the seesaw down the stretch in what became an epic finish.
Jones connected with Stills for a 7-yard touchdown hookup with 4:10 left to give OU a 44-43 lead, but the Mountaineers quickly responded over the next 1:17 and seized the lead back with Bailey's fourth and final touchdown, a 40-yarder from Smith inside of three minutes.
The Sooners got on the board early and often, starting with an 11-play, 70-yard drive capped off by a 4-yard touchdown toss from Jones to fullback Trey Millard that made it 7-0.
Place kicker Michael Hunnicutt made it 10-0 with his 32-yard field goal later in the first quarter.
But that's when the West Virginia offense came to life.
OU seemed to be taking control from that point.
Jones quickly connected for a 76-yard touchdown pass to elusive slot receiver Jalen Saunders and a 4-yard touchdown hookup with Stills moments later.
OU then led 24-10.
West Virginia came back with a score, as Smith hit Bailey for a 33-yard touchdown pass.
Still, the Sooners seemed to be completely in the driver's seat when running back Damien Williams busted loose 48 yards to the house and gave OU a 31-17 advantage.
Hunnicutt had a chance to put the Sooners up 17 at the break but pushed the field goal wide right.
The Mountaineers came out of the second half gates with the first score when Austin sprinted 74 yards to pay dirt.
Jones, who moved into fifth in NCAA history and second in all-time in the Big 12 with his 14,978 yards, tied a program record with his six touchdown passes while going 38-for-51 for 554 yards, despite throwing an interception.
Stills was one of a pair of receivers with four touchdown catches, as he hauled in 10 receptions for 91 yards.
Saunders had 123 yards and a touchdown on seven catches, while wide receiver Justin Brown finished with six receptions and 112 yards.
Williams, who recorded the one touchdown, rushed 22 times for 92 yards.
For West Virginia, Smith finished 20-of-35 for 320 yards and four touchdowns, but he threw a pair of picks.
Bailey amassed 205 receiving yards on 13 catches in becoming the second player to finish with four touchdown grabs.
In leading the Mountaineer rushing attack, Austin averaged an incredible 16.4 yards per carry.
The Sooners will move on to host the Oklahoma State Cowboys at 2:30 p.m. Saturday in a Bedlam game that now looms very large.
A Kansas State loss to Texas and a pair of Sooner victories down the stretch would give OU the Big 12 Championship and automatic BCS bid.