Foster ran for 152 yards and a touchdown, and the Texans are 5-0 for the first time and defeated the New York Jets for the first time in team history, hanging on for a 23-17 victory Monday night.
"You got to give yourself a chance to win and we did that tonight," Sanchez said.
The Texans (5-0) joined the Atlanta Falcons as the NFL's only teams without a loss.
"Arian stepped up and played huge," quarterback Matt Schaub said. "We're very excited to be 5-0 and get a win on the road."
The Jets (2-3) were heavy underdogs after coming off a 34-0 loss to San Francisco last week, but made a game of it. Backup quarterback Tim Tebow got the crowd fired up at times throughout the game, including running for a first down on a fake punt, but had little impact otherwise.
Houston overcame a 100-yard kickoff return by Joe McKnight that cut the Jets' deficit to 20-14.
Schaub was 14 of 28 for 209 yards and a touchdown to Owen Daniels, winning his ninth straight game -- the longest active streak in the NFL -- dating to last season. Daniels finished with four catches for 79 yards for the Texans, who have just their second five-game winning streak in team history.
"It's definitely our best start," Daniels said, "and a big part of that is it doesn't matter on this team about stats or individually what is going on. It's great to be a part of something like that."
The enthusiasm was tempered, though, by a left knee injury to star linebacker Brian Cushing that coach Gary Kubiak said he was "very concerned" about.
"I don't know what happened," Kubiak said. "They tell me he cut on a play and his foot might have got stuck in the turf."
Nick Folk's 21-yard field goal with 11:55 left made it 23-17, but the Jets appeared as though they would get more.
"I honestly thought we were going to drive down the field and score a touchdown" on the last drive," wide receiver Chaz Schilens said. "It's not going the right way."
Sanchez hit Cumberland for 24 yards on first down, and then Jeremy Kerley for 36 yards two plays later. Then, Tebow came in and took the direct snap -- with Sanchez lined up as a wide receiver -- and rumbled up the middle for 13 yards to make it first-and-goal from the 3. The crowd at MetLife Stadium went wild, sensing a chance at Tebow's first touchdown since being acquired by New York in March.
Tebow stayed in but was stopped for no gain. Then, Sanchez -- who has been under fire with many calling for Tebow to take his starting job -- had a pass tipped by J.J. Watt and it was incomplete to Schilens in the end zone on third down.
Foster walked untouched into the end zone for a 13-yard touchdown to give the Texans a 14-7 lead with 11:11 left in the first half. He gained 42 yards on three straight carries on the 67-yard drive, and had 100 yards on the ground at halftime on just nine carries.
The Jets threatened late in the half, but Sanchez's pass was tipped by Watt and went off Kerley to Brice McCain, who ran back the interception 86 yards before Kerley caught him at the New York 9.
Shayne Graham made a 27-yard field goal as the half ended to make it 17-7. He added a 42-yarder late in the third quarter to make it 20-7.
McKnight got the Jets right back into it with his 100-yard kickoff return -- his second of his career -- to cut the Texans' lead to 20-14 with 4:38 left in the third quarter. McKnight sped up the middle, cut to his left and ran out of a shoulder tackle attempt by DeVier Posey and dived into the end zone.
New York attempted an onside kick, but the ball went through Schilens' hands and Shiloh Keo recovered, giving the Texans the ball with good field position.
Schaub hit fullback James Casey for 30 yards to quiet the crowd, but the Jets' defense held the Texans to a 22-yard field goal with 13 seconds left in the quarter.
The Texans opened the game by scoring on an efficient 8-play, 85-yard drive capped by a 34-yard touchdown catch by a wide-open Daniels less than 4 minutes in. Daniels, who earlier gained 14 yards on a screen pass, got wide open behind the secondary as New York's safeties both bit on a play-action fake by Schaub.
New York then went three-and-out, an all-too-common series for the Jets this season -- as chants of "Tee-boww! Tee-boww!" started after Sanchez's first incomplete pass on second down. Tebow made his first appearance on the Jets' fifth play from scrimmage and, to a wave of cheers, threw a pinpoint deep pass to newcomer Jason Hill. But the receiver dropped it, Tebow departed and Sanchez missed Clyde Gates on third down, bringing on a chorus of boos.
Jets coach Rex Ryan promised his defense would be sturdier than in last week's rout at the hands of San Francisco, and after Houston's scoring drive, it was. Cromartie stepped in front of Andre Johnson and picked off Schaub's poorly thrown pass on the Texans' third possession, then gave the ball to his wife in the stands.
Four plays later, the Jets tied it for their first points in two weeks -- a 27-yard TD catch by Cumberland.
Sanchez zipped a pretty pass to Gates on first down for a 27-yard gain on the scoring drive. After a 2-yard run by Bilal Powell and McKnight got 1 yard, Sanchez threw yet another terrific toss to Cumberland, who caught the ball over Quinn, bobbled it and tucked it in as he fell to the turf in the end zone to tie it at 7 with 4:07 left in the opening quarter.
Tebow got the crowd going when the Jets went for it on fourth-and-1 at their 24 on the first play of the second quarter. The do-it-all backup quarterback took the snap and ran off left tackle for 3 yards. But the drive fizzled with three straight incompletions by Sanchez.
Cromartie held Johnson to one catch for 15 yards and Johnson also had a drop. ... Jets C Nick Mangold left early in the third quarter with a left ankle injury, but returned later in the period. Jets backup receiver Clyde Gates injured his left shoulder and his arm was in a sling after the game. Linebacker Josh Mauga had his right arm in a sling afterward. Jets defensive tackle Kenrick Ellis injured in a knee in the fourth quarter.
The NFL says it is close but still has yet to interview the four players named along with Peyton Manning in a report five months ago that linked them to PEDs.
ESPNDallas columnist Jean-Jacques Taylor and NFL Nation reporter Todd Archer discuss what Randy Gregory needs to do to have a long future with the Cowboys, plus a Dez Bryant-Russell Westbrook comparison.
ESPN Bears reporter Jeff Dickerson discusses how Kevin White looked in Chicago's OTA on Wednesday.
ESPN Redskins reporter John Keim says Preston Smith and the defense, including safety DeAngelo Hall, had a big day against the offense Wednesday.
ESPN Browns reporter Pat McManamon explains that perennial Pro Bowl tackle knows the wear and tear on his body; if he needs a day off in OTAs and during the season, Cleveland Browns are wise to give it to him.
DeSean Jackson has missed too many voluntary workouts to collect his $500,000 bonus, but Washington isn't concerned about his absences.