Foles injured his left, non-throwing shoulder on the last play of the first quarter when he was sacked by Texans linebacker Whitney Mercilus.
Mark Sanchez, who played three quarters Sunday in his first regular-season appearance since 2012, will fill in for Foles.
Foles had highs and lows before departing Sunday. He threw a perfect pass to Jeremy Maclin for a 59-yard touchdown, but a few minutes later, his pass for Riley Cooper was intercepted and returned 51 yards for a touchdown by Texans cornerback A.J. Bouye.
As the quarter ended, Mercilus sacked Foles at midfield. He was slow to get up and walked off the field with a trainer before heading to the locker room for X-rays on his shoulder and collarbone.
Foles did not return to the contest, opening the door for Sanchez, who completed 15 of 22 passes for 202 yards, two touchdowns and two interceptions in a 31-21 victory.
"I thought Mark did a really good job," Kelly said. "The one interception -- I think it hit wide receiver Josh Huff in the hands and was tipped. The other one, I think [Cooper] got spun around. Besides that, I thought he had a really good command of what we're doing.
"Since day one here, you know what type of professional [Sanchez] is. ... We were fortunate when Nick did go down, we were able to bring Mark in. I don't think we missed a beat."
Information from ESPN.com Eagles reporter Phil Sheridan contributed to this report.