Foster scored on an 8-yard run in the first period and capped the game's scoring with a tackle-shedding, 42-yard scoring scamper in the fourth as the Texans advanced in the first post-season appearance in franchise history.
J.J. Watt returned an interception 29 yards for a score just before the intermission for the Texans, who snapped a three-game losing streak and now move on to face Baltimore in next week's AFC semifinals.
Andy Dalton went 27-of-42 through the air for 257 yards but didn't throw a touchdown and was picked off three times for Cincinnati.
The Bengals failed in their bid to log their first playoff win since 1991.
Celebrity Families of 2014 [PHOTOS]