Arizona (10-5) led 2-0 after the first and also had a 3-2 edge after four but Houston scored single runs in the fifth and sixth to go up 4-3.
The Wildcats tied it in the bottom of the sixth on an RBI single by Dayton Dooney before breaking it open in the seventh.
Branden Boissiere's run-scoring single started the rally, then Arizona's next four runs came in either via walks or hit batters with the bases loaded to make it 9-4.
Boissiere had three hits and three RBI while Mathew Dyer and Mac Bingham had two hits apiece.
Peyton Price picked up the win in relief.
Arizona moves on to Pac-12 play next weekend, hosting Oregon State for three games. Before that it will visit ASU on Tuesday night in a non-conference tilt.