Pickett, who threw for a school-record 455 yards last week, completed just 18-of-36 passes for 252 yards Saturday, but could not be stopped by Arizona State when the game was on the line.
His late heroics allowed John Anderson to kick a 30-yard field goal as time expired.
Pickett converted two key third-down opportunities on the final drive, which spanned the final 7:12. He rushed for a first down on a 3rd-and-8 and successfully executed an option for another first down on 3rd-and-9.
The sophomore quarterback brought the Huskies to the 11 before Willie Hurst was stuffed by the Sun Devils' defense on 3rd-and-1. Washington set up for a 28-yard field goal with 51 seconds left, but Arizona State was called offsides, giving the Huskies a first down.
Washington (6-1, 4-1 Pac-10) let the clock tick down to three seconds before Anderson kicked his winning field goal to give the Huskies their fifth fourth-quarter comeback victory of the season.
Washington is undefeated in its last 11 games decided by seven points or less.
Hurst rushed for 181 yards on 21 carries and scored a touchdown for Washington, which squandered a 24-7 second-quarter lead.
Arizona State (4-3, 1-3 Pac-10) scored 19 straight points bridging the second and third quarters, led by quarterback Jeff Krohn, who completed 16-of-24 passes for 290 yards and threw two touchdowns.
Shaun McDonald caught seven passes for 154 yards, including a 23-yard touchdown from Krohn that gave the Sun Devils their first lead, 25-24, in the third quarter.
Delvon Flowers, who rushed for a career-best 226 yards last week, was limited to 69 yards on 20 carries for Arizona State.