Keith Price completed 16-of-21 passes for 112 yards and a touchdown with an interception for the Huskies (9-4), who also received a 100-yard kickoff return touchdown from John Ross.
"Bishop's done a great job of carrying us all season," Price said of Sankey. "He's a workhorse."
Washington, which secured its first nine-win season since 2000, was coached by quarterbacks coach Marques Tuiasosopo after former bench boss Steve Sarkisian jetted to USC. The school replaced Sarkisian with former Boise State coach Chris Petersen, but decided to go with Tuiasosopo for the bowl game.
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Taysom Hill did just about everything for BYU (8-5), rushing for 143 yards and a score on 29 carries while completing 22-of-40 passes for 274 yards with an interception.
Cody Hoffman hauled in a game-high 12 passes for 167 yards for the Cougars, who have appeared in a bowl in each of head coach Bronco Mendenhall's nine seasons on the sidelines.
"I told my team I was proud of them," Mendenhall said. "I like how hard they played."
Sankey got the Huskies on the board on their first touch of the game, capping a 12-play, 71-yard trek with an 11-yard touchdown run off the right edge to make it 7-0 just over six minutes into the game.
Sankey was stuffed for a 9-yard loss on 4th-and-1 from the BYU 3 late in the opening frame, and Hill led the Cougars down the length of the field on the team's ensuing touch to tie the tilt.
Hoffman had three catches on the march, including a 22-yard grab down to the Washington 13. Three plays later, Hill surged into the end zone from a yard out to bring BYU even.
However, Ross broke free down the right sideline and outran everyone en route to a 100-yard kickoff return TD that made it 14-7 Washington.
Justin Sorensen sandwiched a pair of field goals around a Washington three- and-out to get the Cougars within 14-13, but following a 47-yard kickoff return from Jesse Callier, Sankey scored from 11 yards out to put Washington on top, 21-13, with 2:36 left in the half.
Sorensen's 32-yard kick sent BYU into the break facing a five-point deficit.
Washington extended the lead on its opening touch of the second half, as Price broke free for 20 yards on 3rd-and-8 from the BYU 49 before capping the 10- play series with a 16-yard scoring strike to Austin Seferian-Jenkins that made it 28-16 with 11:12 left in the third.
Robertson Daniel picked off a Price pass late in the quarter to set the Cougars up at the Washington 26, but Sorensen missed a 44-yard try to keep the difference at 12.
Travis Coons kicked a 45-yard field goal midway through the fourth to extend Washington's lead and John Timu's interception of Hill with under three minutes left all but sealed the outcome.
Ross' 100-yard kickoff return TD was the second in Washington history. The other was by Paul Arnold in 1999 against Air Force ... Washington made its fourth straight bowl appearance ... BYU racked up 28 first downs, compared to 19 for Washington ... The Cougars compiled 473 yards of offense and held the Huskies to 316 yards.
12/28 02:00:38 ET