Foster carried the ball 19 times, Tate rushed 12 times and each scored a touchdown for the Texans (6-3), who reached three games above .500 for the first time in the franchise's nine-year history.
The Texans won despite the absence of star receiver Andre Johnson for the third straight game.
Matt Schaub completed 14-of-23 passes for 119 yards with an interception.
Cleveland (3-5) could manage only 172 yards of total offense against Houston's defense. The Browns were playing without running backs Montario Hardesty and Peyton Hillis, who were both out with injuries.
Colt McCoy went 14-of-22 through the air for 146 yards with a touchdown and an interception.