Zorn said Betts will have an MRI exam on Monday to determine the severity of the tear.
Adding to the injury concerns was the announcement by Zorn that Clinton Portis, out since Nov. 8 because of a concussion, will not play next week against the Philadelphia Eagles.
Washington has shuffled its lineup throughout the season because of multiple injuries to key players.
Left tackle Chris Samuels, right guard Randy Thomas and safety Chris Horton are on injured reserve.
Tight end Chris Cooley (right ankle surgery) missed his third straight game and defensive tackle Albert Haynesworth was inactive because of a sprained left ankle sustained last week.
"Everyone is trying to stay positive," Washington quarterback Jason Campbell said. "Even though we've lost a lot of guys, no one got down on themselves."
Betts' injury leaves Washington even thinner at running back. With Portis also hurt, Rock Cartwright and Quinton Ganther are the backfield options for now.
Portis visited a specialist in Pittsburgh on Friday and hasn't been able to practice since he was injured on a helmet-to-helmet hit during a to the Atlanta Falcons.
Betts has been filling in for Portis and apparently got hurt on a hard tackle by Bradie James at the end of a 2-yard gain. He finished with 5 yards on four carries.
Without their top two runners, Cartwright rushed for 67 yards and caught seven passes for 73 yards. He had only 17 carries coming into the game.
"I have an opportunity," Cartwright said. "All I can do is take advantage of the workload and play on Sundays."
"We've got a big game next week," Zorn said. "We're going to look at our roster, patch it up and go again."
The Associated Press contributed to this report.