Redskins coach Jay Gruden did not offer a timetable for Griffin's return, announcing that the third-year quarterback will undergo an MRI on Monday to determine the extent of the injury. He likely would miss the rest of the season if the medical tests determined that he needs surgery.
"I know he's sick about it, and I'm sick about it for him," Gruden said. "But he will get well, and he will get well as soon as he can. ... We're sick about it, but now we have to move forward."
Griffin, 24, said he is trying to stay positive despite the potential of a season-ending injury.
"I'm just trying to stay in a good place," he said. "Our spirits are high -- we're always thinking positive. We'll see what happens tomorrow, and whatever information comes out, we'll know what the timetable is from there."
Griffin suffered the injury while making an acrobatic, leaping, across-the-body throw near the sideline while being pursued by linebacker Paul Posluszny. Griffin landed awkwardly on his ankle as he finished the throw, then tumbled out of bounds.
"You never want to get hurt," Griffin said. "You want to be out there and be available for your teammates. It's just an unfortunate situation because it wasn't on a hit or anything. It's just something that happened.
"In moments like these, you've just got to keep the faith, stay with it and just believe that everything happens for a reason."
While play continued, Griffin's leg was placed in a splint. He wiped his brow and waved, saluted and gave a thumbs-up signal to the crowd as he was carted along the sideline toward the players' tunnel.
"It's an excruciating pain, but I think anytime you look at your ankle and it's in a different direction, you're probably not going to try to get up," he said. "I was just trying to make sure I wasn't doing any more damage to it, so I just stayed there and waited for the trainers to come and help me get up and on the cart."
It was the latest setback in what has been a promising but injury-marred career for Griffin, who missed all or part of four games in his record-setting rookie season of 2012 due to various injuries, including a torn right ACL that led to reconstructive knee surgery.
Griffin, who also tore his right ACL in college at Baylor, was replaced by Kirk Cousins, who passed for 250 yards and two touchdowns in Washington's 41-10 victory.
"Things happen in pro football," Gruden said. "Starters get hurt, and backups have to come in. This is an extreme circumstance, but Kirk is more than ready to take over."
Griffin also expressed confidence in Cousins.
"I believe that Kirk is going to go out there and play extremely well," he said. "This team is going to win games because we have a very good team that's on that field. One guy doesn't define this team -- that includes me."
Redskins receiver DeSean Jackson also left the game in the first quarter with a shoulder injury and did not return. He was diagnosed with an AC joint sprain and is considered day-to-day.
"This one was a little vicious, the pain right on the bone and things like that," Jackson said. "So I'm day-to-day."
ESPN.com Redskins reporter John Keim and The Associated Press contributed to this report.