Haws hit 15-of-24 shots and sank six free throws in the win, which set up a second-round meeting with the winner of Wednesday's game between Tennessee and Mercer.
Brandon Davis poured in 22 points for BYU (22-11) and Matt Carlino tallied 20 points and nine assists.
Sixth-seeded Washington (18-16) had four players in double figures, led by C.J. Wilcox with 20 points. Shawn Kemp notched a double-double with 15 points and a game-high 11 rebounds.
The game featured seven ties and eight lead changes. The Cougars outscored Washington 57-44 in the second half.