West Virginia sacked Tino Sunseri 10 times, including four on Pittsburgh's final drive, when time ran out on the Panthers.
The Mountaineers (8-3, 4-2 Big East) can earn a share of the conference title next week at South Florida. West Virginia also holds slim hopes for earning the league's automatic BCS berth and would need some help in a tight league race.
Pittsburgh (5-6, 3-3) led 17-7 at halftime but was limited to 80 yards in the second half.
Pittsburgh starting running back Zach Brown left the game with an undisclosed injury just before halftime and backup Isaac Bennett left the game midway through the third. Bennett would later return, but Pittsburgh's offense stalled and West Virginia gained momentum.
Alston's 8-yard TD run midway through the third quarter cut West Virginia's deficit to 20-14.
In the fourth quarter, the Mountaineers turned the ball over on downs and Tyler Urban fumbled the ball away on the Mountaineers' next drive.
But a personal foul on Pittsburgh's Aaron Donald gave West Virginia a first down at the Panthers' 28. On fourth-and-7, Geno Smith hit Tavon Austin for nine yards. Alston followed with an 11-yard run and then ran in the go-ahead score to cap the 83-yard drive and give West Virginia its first lead of the game.
Pittsburgh was forced to punt with 2:30 left and got one last chance from its own 34 after West Virginia went three-and-out. Sunseri ran for a first down on fourth-and-1. But he was called for intentional grounding on the next play and Pittsburgh couldn't move the ball as time ran out.
Sunseri went 12 of 23 for 137 yards.
Mistakes on West Virginia's special teams contributed to Pittsburgh's first 17 points.
West Virginia's George Wright was called for a low block on a Pittsburgh field goal attempt that went wide right in the first quarter. Brown scored from a yard out five plays later.
Bad punting has been a problem all season for the Mountaineers. Freshman Michael Molinari's 22-yarder late in the first quarter set up Pittsburgh near midfield, and the Panthers scored in five plays, with Bennett going in from 6 yards out for a 14-0 lead.
Molinari's next punt went 27 yards and he was replaced by Corey Smith, who was stellar the rest of the game. West Virginia punted on six of its eight first-half possessions, went 0-for-6 on third down and was held to 141 total yards before halftime.
A bouncing Pitt punt went off West Virginia's Ishmael Banks and the Panthers recovered at the West Virginia 33, leading to a field goal for a 17-7 halftime lead.
The special teams follies continued in the third quarter when Austin fumbled a punt and Pittsburgh recovered at the Mountaineers 16, leading to another Kevin Harper field goal.
West Virginia replaced two of its offensive linemen early in the third quarter. And despite being held to minus-2 yards rushing in the first half, the Mountaineers stayed with their ground game. West Virginia ran for 44 yards on its second drive of the third quarter that led to Alston's second score.
Smith completed 22 of 31 passes for 244 yards and set school single-season school records for pass completions (291), attempts (448) and yards (3,741), formerly held by West Virginia quarterback Marc Bulger in 1998.
Midway through the second quarter, West Virginia's Stedman Bailey caught a long pass over the middle and appeared to be hemmed in by three defenders, but he reversed course and scored on a 63-yard pass play.
Bailey caught three passes for 80 yards, giving him a school-record 1,117 yards, breaking the old mark of 1,043 set by David Saunders in 1996. Austin's 10 catches for 102 yards gave him 82 receptions, breaking the mark of 77 shared by two others. He now has 1,009 yards on the season, giving the Mountaineers two 1,000-yard receivers for the first time in one season.