Anthony Washington scored 10 points for the Huskies (4-4).
Louis Truscott recorded his fifth double-double of the season with 21 points and 11 rebounds, while Andre Owens added 13 points and Bryan Shelton 10 for the Cougars (2-6).
The Huskies used full-court pressure to set a season-high for steals before halftime and Houston committed a season-worst 24 turnovers.
Washington jumped out to an 11-2 lead that included seven points off turnovers and led 20-9 after a three-point play by Allen.
Truscott and Owens scored five points apiece as Houston responded with a 12-4 run to close to within 24-21 with five minutes remaining in the first half.
Two 6-0 runs late in the half put the Huskies up 44-28 at halftime, their largest halftime lead of the season.
Houston turned the ball over three times before scoring in the second half and the Huskies led by at least 17 points from that point.