Sources told NFL Network that Burrow tore the ACL in his left knee after he was hit by two Washington defenders in the third quarter of the loss at FedEx Field in Landover, Md.
Burrow was carted off the field and did not return to the game.
"Thanks for all the love," Burrow tweeted Sunday night. "Can't get rid of me that easy. See ya next year."
Bengals coach Zac Taylor did not provide specifics about Burrow's injury when he spoke to reporters after the game. Burrow is expected to get an MRI on Monday to confirm the injury.
Burrow, the No. 1 overall pick out of LSU in April's NFL Draft, completed 22 of 34 passes for 203 yards and a touchdown before he left Sunday's game.
He completed 65.3% of his throws for 2,688 yards, 13 touchdowns and five interceptions in 10 games this season and was a contender for AFC Offensive Rookie of the Year. He posted a 2-7-1 record as a starter in his rookie campaign.
The Bengals will now turn to backup quarterback Ryan Finley for the rest of the season. Finley completed 3 of 10 passes for 30 yards and an interception on Sunday after replacing Burrow.
"To lose your quarterback is tough, but that's the way it goes," Taylor told reporters. "Finley came up and we have to help him. I have to put him in better situations."