The Texans will next play at New England on Jan. 13.
Foster became the first player in NFL history to have 100-yard games in each of his first three playoff games.
The Texans (13-4) had trouble finishing drives all day and mustered only three field goals in the first half. Houston struck first after the break, with Foster scoring the game's only offensive touchdown on a 1-yard run in the third quarter to make it 16-7.
Johnathan Joseph had an interception and J.J. Watt had a sack as the Texans beat the Bengals (10-7) in the wild-card round for the second straight year. Cincinnati hasn't won a playoff game since 1991, the league's longest current streak.