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.
Toddler uninjured after being knocked over by Obama family dog
Ron Burgundy interviews Peyton Manning on SportsCenter