Avery, who led Rams wide receivers with 47 catches last year, was hurt with 1:32 left in the first half. He jumped for Sam Bradford's long pass down the right sideline, fell and clutched his knee as the ball dropped incomplete.
"He probably has a significant knee injury," St. Louis coach Steve Spagnuolo said. "We'll know further when we get the test. Hopefully, it's not."
Before Avery was carted off the field, Patriots wide receiver Wes Welker went over and extended his hand. Welker is playing again after having surgery for a torn ligament in his left knee in February.
Avery, a second-round draft pick out of Houston in 2008, had 53 receptions as a rookie and started 28 games in his first two seasons. He caught two passes for 48 yards Thursday before getting injured.