Foster carried the ball 29 times and found the end zone on a 13-yard run in the second quarter that put the Texans ahead to stay at 14-7.
Matt Schaub completed 14-of-28 passes for 209 yards with a TD and an interception as Houston moved to 5-0 for the first time in franchise history.
Defensive end J.J. Watt had six tackles and a sack while Kareem Jackson intercepted a Mark Sanchez pass late in the game to seal the Texans' victory.
Owen Daniels made four grabs for 79 and a score for the victors.
Sanchez hurled for 230 yards but was picked off twice for the Jets (2-3), who dropped their second straight.
Jeremy Kerley led the New York receiving corps with five grabs for 94 yards.