Harris, a Shorecrest High School grad, nodded in James Moberg's corner in the 15th minute for the game's lone goal. The tally came right after UW forward Nate Sackeyfio put a header off the post.
Husky goalkeeper Spencer Richey needed to make just one save to earn his 10th shutout of the season.
The Falcons' had an opportunity to tie the match when a second-half free kick hit the crossbar and a follow-up shot missed the net.
Washington (13-4-3) advances to play at No. 12 seed Creighton on Sunday at 3 p.m.
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