Texans kicker Matt Ammendola kicked a 45-yard field goal with 10 seconds left in the first half to give Houston a 10-7 halftime lead on the Bengals.
Ammendola was signed this week to replace injured kicker Ka'imi Fairbairn and was only 2-of-6 in his career from 40-49 yards before Sunday.
Cincinnati has only 109 yards against a team decimated by injuries this week. Houston ruled out nine players on Friday.
Burrow is 14-of-18 for 94 yards and the touchdown.
The Texans have 273 yards and could have a bigger lead if not for two C.J. Stroud fumbles, one on fourth down with the Texans at the Cincinnati 9 and another on a hit by Mike Hilton strip-sack of Stroud with Houston at the Cincinnati 31.
Stroud stayed on the ground briefly but did not miss a play.