Burrows had successful surgery Monday and is expected to miss four weeks.
Burrows sustained the injury when he was hit by defenseman Christopher Tanev's clearing attempt. He stayed in the game and wound up playing 18:47 over 29 shifts.
"Alex is a big part of our team and it's a big loss, but we'll come together and he'll be back soon," goaltender Roberto Luongo said.
Burrows has gone 22 straight games without a goal dating to last season. He missed 12 games in October with a broken foot, an injury he sustained in the season opener.
He has three points and a minus-7 rating in 17 games this season.