Thomas, who led the Cougars (9-5, 1-0 AAC) with 23 points and eight rebounds, converted two shots from the line with 9.2 seconds to play and seal the win in Houston's first game in the American Athletic Conference.
Thomas also made a key block against UConn's (11-2, 0-1) Shabazz Napier in the closing seconds to thwart the Huskies, who have lost two of their last four games.
Jherros Stiggers added 17 points off the Houston bench and Danrad Knowles scored 13.
Napier led UConn with 27 points and nine rebounds. Ryan Boatright tallied 17 points.
Houston took a 46-28 lead at halftime and weathered the second half, which the Huskies dominated 43-29.