Young went down while scrambling and immediately grabbed his left leg. He missed a victory against Philadelphia last week with a sprained left knee and ankle.
"I reinjured it," said Young, who was in a walking boot as he left the locker room afterward.
"It's very tough," he said. "I definitely want to be out there with my teammates. It's very frustrating. I hate putting them in that situation, but we have an outstanding backup quarterback."
Young was replaced by Kerry Collins.
The Titans, who are off next Sunday, lost 33-25.
"The bye is good, to get back to rest a little bit and get more treatment and get off of it and let it rest," Young said.
The Titans also lost wide receiver Kenny Britt to a hamstring injury in the first quarter.
The second-year pro had a career day against the Eagles last week with seven catches for 225 yards, including touchdowns of 26, 80 and 16 yards. It was the best performance by a receiver in the NFL this season.
Information from The Associated Press contributed to this report.