"We willed that win," WVU coach John Beilein said.
Sally broke a 58-all tie with eight straight points for a 66-58 lead with three minutes left. But West Virginia (14-7, 4-6 Big East) then turned the ball over three straight times.
Providence (10-13, 0-9) capitalized, hitting a 3-pointer and feeding a dominating Ryan Gomes inside twice to pull within 66-65.
But Big East leading three-point shooter Donnie McGrath missed a wide open look from the top of the key, and D'or Fischer made the play of the game by blocking Gomes' shot with 16 seconds left.
"Gomes is a high-profile player who I wanted to block, so I made sure I didn't make any contact," said Fischer, whose three blocks moved him to 18th on the all-time NCAA list.
Sally, who scored the last nine points of the game, was fouled and hit one free throw for the final 67-65 edge before Parmer's 3-pointer banged off the front rim at the buzzer.
Sally's shot was WVU's 21st free throw. Providence shot just three all game.
"I've never had a team shoot three free throws in a game, ever," Providence coach Tim Welsh said. "I mean, my junior high team did once. We had a bunch of skinny, weak kids. But that's the first time I have ever seen that in a Big East game. We have the best post player in the league and can't get to the foul line. That might be a record."
Welsh's frustration stems from Providence's 12th straight Big East loss and eighth in nine games. It was their fifth loss by five points or less this year.
The Friars are off to the worst conference start in school history and the worst mark overall since they lost all six of its games in 1979, the Big East's first year.
The win kept alive slim WVU NCAA hopes. The Mountaineers, winners of two straight Big East games for the first time this year, will likely have to win five of their last six to sneak in.
Only Gomes' Big East-leading 40th career double-double allowed Providence to keep pace. He had 14 rebounds and scored a career-high 33 points, including 20 of his teams 26 in one stretch.
His 16 made field goals are the most for a Big East player this year.
"Gomes is absolutely incredible," Beilein said. "He is an incredible talent and we were fortunate to come out with a win with no answer for him."
It was Gomes' second consecutive career-high scoring game. He had 32 versus Villanova on Saturday.
McGrath also came up just short, missing a chance to tie an NCAA record with 15 straight 3-pointers. McGrath had hit 11 consecutive 3-pointers entering the game.
He made 3s on three straight possessions versus WVU, but missed his fourth try – this from the left side with 2:03 left in the first half.
McGrath finished four of nine and scored 14 points. He is 26 of 38 (68.4 percent) in his last six games.
Kevin Pittsnogle, in his just third start of the year, scored 13 points. Frank Young added a career-high 11.
The game was played on ESPN Classic as part of ESPN's 25th-year celebration. Fans danced to songs and wore clothing from the 1970s and 80s. The best-dressed won an Atari gaming system.