The Rockets were scheduled to leave after the game for two games in China against the New Jersey Nets on Wednesday and Saturday.
Trailing 72-67 to start the fourth quarter, the Cavaliers went on a 21-8 run and took an 88-80 lead with 6:10 to play. Manny Harris had eight points in the Cleveland surge and the Rockets never regained the lead.
Cleveland led 89-86 when Eyenga hit consecutive 3-pointers late in the fourth quarter to pad the Cavaliers' lead.
Ramon Sessions had 15 points for Cleveland. Harris finished with 14 points.
The Cavaliers took a 47-45 halftime lead, but the Rockets rallied in the third quarter for a 72-67 lead going into the final period.
Budinger helped out with 11 straight Rockets points in the quarter and rookie first-round pick Patrick Patterson hit two baskets in the final minute to keep Houston ahead.
Neither team could get a clear advantage in the first half. The Cavaliers took a 47-45 lead at intermission on a short jumper by Sessions in the final second.
Hickson led Cleveland with 16 first-half points, including eight over the final 2:44 of the second period.
Backups played much of the first quarter and the Rockets emerged with a 23-19 edge.
Mo Williams (right groin) and Loren Woods (left Achilles tendinitis) sat out for the Cavaliers. ... Yao Ming sat out as part of his recovery from foot surgery that caused him to miss all of last season. Yao played 12 minutes in each of the Rockets first three preseason games. He scored 10 points in Saturday's 126-92 victory over the Indiana Pacers.