Williams caught seven passes for 160 yards as the Huskies tried to put a last-second loss to Michigan last week behind them.
Washington began a five-game homestand, its longest since playing its first six here in 1931.
San Jose State (1-1) took a 10-0 halftime lead on a one-yard run run by Osmar Staples and a 43-yard field goal by Nick Gilliam.
Washington (1-1) took control after intermission, beginning with a 22-yard touchdown pass from Pickett to Wilbur Hooks. Pickett connected on a 21-yard pass to Williams 13:06 later to put Washington in front to stay at 14-10.
Pickett gave the Huskies a 24-10 cushion by connecting on the 89-yarder with Williams 15 seconds into the final period.
The defense added a score when quarterback Scott Rislov, who was trying to avoid a penalty for intentional grounding, had a pass picked off by tackle Terry Johnson in the end zone early in the final quarter.
John Anderson added field goals of 29 yards in the third quarter and 31 yards in the fourth.
Washington has won all eight games against San Jose State with every game taking place in Seattle.