Skowron, who also had an 18-yard touchdown run on a fake field goal, missed a first attempt wide right but the Huskies called timeout just before the kick and he got a second try.
The kicker gained redemption after missing the go-ahead 36-yarder in the final minute of a loss to USC.
Washington (6-5, 2-5 Pac-12) was running out the clock when Tra'Mayne Bondurant knocked the ball out of running back Deontae Cooper's hands and Derrick Turituri recovered for Arizona (8-2, 5-2 Pac-12, No. 14 CFP) at the Washington 45 with 1:23 to go.
The Wildcats moved to the 30 and left it up to Skowron.
Just before the winning field goal, Anu Solomon threw 34 yards to Caleb Jones, who made a tough catch in the back of the end zone for what initially was ruled an Arizona touchdown.
But replays showed Jones' toe was on the out of bounds line and the TD was negated.
Washington amassed 504 yards to the Wildcats' 375.
But Cooper's turnover was the Huskies' third lost fumble of the game.
Quarterback Cyler Miles fumbled at the Arizona 14 in the first half, then lost another that led to a Wildcats touchdown in the crazy final two minutes of the first half.
Trailing 17-7, Arizona set up for a field goal try but Skowron took a direct snap and ran around the right side right to the end zone, an 18-yard touchdown that cut the lead to 17-14 with 1:39 left in the half.
On Washington's next possession, Miles couldn't handle a long snap, picked the ball up, then fumbled and Bondurant recovered for Arizona at the Huskies 19 with 33 seconds left in the half. Three plays later, Nick Wilson ran 8 yards up the middle for a touchdown that gave the Wildcats a 21-17 lead eight seconds before halftime.
The Huskies pushed the Wildcats around much of the game, but were undone by turnovers and penalties. Washington was penalized 13 times for 111 yards to Arizona's two penalties for 14 yards.
Dwayne Washington rushed 13 times for a career-high 148 yards for the Huskies. He burst through the line for a 66-yard touchdown pass in the first half. And his 1-yard TD put Washington up 26-21 with 2:45 left in the third quarter. That run capped a 13-play, 74-yard drive.
Solomon's 63-yard pass to Trey Griffey set up Skowron's 35-yard field goal that cut the lead to 26-24 with 13:08 remaining.
Washington had a 347-193 advantage in total yards in the first half but trailed at the break, thanks largely to nine penalties for 67 yards.
Cameron Van Winkle's 43-yard field goal attempt for the Huskies bounced off the right upright and was no good.
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