Foster had 19 carries for 124 yards and a score, while Tate added 115 yards and a touchdown on 12 rushes for Houston (6-3), which is three games over .500 for the first time in franchise history.
Johnson missed his fifth consecutive game since suffering a right hamstring injury during the Texans' Week 4 game against Pittsburgh.
Matt Schaub threw for 119 yards and an interception on 14-of-23 passing, while Houston's defense limited the Browns to just 172 yards of total offense.
Colt McCoy was 14-of-22 for 146 yards with a touchdown and an interception for the Browns (3-5), who have dropped four of their past five games.
The Texans went 82 yards in nine plays on the opening drive and Tate had a career-long TD run of 27 yards to put Houston ahead 7-0. Schaub was 4-for-4 for 38 yards on the possession with Joel Dressen hauling in two passes for 27 yards.
DeMeco Ryans stripped Ogbonnaya on Cleveland's first offensive play and J.J. Watt recovered the ball for Houston at Cleveland's 28-yard line. Schaub got in from two yards out on the sixth play of the drive as Houston took a quick 14-0 lead midway through the first quarter.
Josh Cribbs took the ensuing kickoff 63 yards down the right sideline to Houston's 37-yard-line, but was called for a facemasking penalty while trying to stiff arm Brice McCain. The Browns settled for three as Phil Dawson's 50- yard field goal attempt split the uprights.
Foster sliced through the middle of the Cleveland defensive line and scampered 19 yards to paydirt to give Houston a 21-3 lead midway through the second quarter. The run gave Foster his 24th career rushing touchdown in Houston, passing Domanik Williams for the franchise record.
With 1:06 to go until halftime, Browns head coach Pat Shurmur elected to go for it on 4th-and-2 at midfield.
McCoy found Evan Moore for a pick up of six yards to move the chains, but Quintin Demps intercepted McCoy two plays later and returned the ball 42 yards to Cleveland's 20-yard line with two seconds left in the half. An unnecessary roughness penalty was called on Cleveland's Jason Pinkston on the run back and Neil Rackers converted a 28-yard field goal to give Houston a 24-3 lead at halftime.
The Texans settled for a 21-yard Rackers field goal on their first possession of the second half to cap off a 13-play, 78-yard drive which took over seven minutes off the clock. Foster carried the ball five times for 40 yards on the march.
Dawson booted one through from 51 yards out to bring Cleveland within 27-6, late in the third quarter.
Tate ran for 25 yards, with Foster adding 20 more on Houston's ensuing drive as Rackers added three more points with a 50-yard field goal.
Cribbs hauled in a two-yard touchdown reception to make it a 30-12 game with 4:47 to go in final quarter. The two-point conversion was unsuccesful.
The Texans have scored 68 points this season in the first quarter, while Cleveland has just six points in the opening stanza this year...Dawson set a career-high with his sixth field goal made over 50 yards this season...Foster posted his sixth straight game with over 100 yards from scrimmage...The 261 yards gained on the ground by the Texans is a new single-game franchise record.