Williams led four players in double figures for Maryland (24-8), which advanced to play Sunday against fifth-seeded Michigan State, a 70-67 winner over New Mexico State.
Aubrey Coleman, the nation's leading scorer, had 26 points for 13th-seeded Houston (19-16). The senior averaged 25.6 points per game this season. Kelvin Lewis added 24 for Houston, 17 in the second half.
Maryland overpowered the Cougars 50-29 on the boards and held them to 41 percent shooting. Houston was also hurt by 20-of-32 shooting from the free throw line.
The Terrapins made 21 of 25 free throws, which helped them keep Houston from making a run in the second half.
Landon Milbourne added 19 points, Vasquez 16 and Eric Hayes 11 for Maryland. The 6-foot-10 Williams made nine of 14 shots.
Maryland has been to the NCAA tournament eight times this century. This was Houston's first appearance since 1992.
With Vasquez held to six points, Maryland still led 43-42 early in the second half. Then the Terrapins went on a 12-2 run, capped by Adrian Bowie's 3-pointer, for a 55-44 lead with 16:45 left.
Lewis replied with five straight points for Houston to close the gap to six. But Maryland worked the lead back to double digits, mostly at the free throw line.
After Zamal Nixon's free throw brought Houston within 72-63, Williams' layup pushed the lead back to double figures with 6:49 remaining.
Houston would not go away. Adam Brown's 3-pointer and three free throws by Lewis cut Maryland's lead to 78-69 with 5 minutes left. But that margin held, as Coleman went 11 minutes without scoring late in the second half.
In the first, Houston held a 24-22 lead despite making just eight of its first 25 shots. Coleman had 11 of those points.
Maryland pulled ahead 27-25 on Bowie's 3-pointer. Williams put back a couple of missed shots as Maryland built a 35-29 lead with 2:27 left in the first. Williams had 10 points and 10 rebounds in the first.
