Burrows, who plays on a line with the Canucks' all-star twins Henrik and Daniel Sedin, had the jaw successfully repaired during a Monday surgery in Nashville, a team official confirmed to The (Vancouver) Province.
The ninth-year pro was hurt in Sunday's 3-2 win over the Carolina Hurricanes when he was struck in the jaw on a clearing attempt teammate by Chris Tanev.
"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," Canucks goaltender Roberto Luongo said.
Burrows also missed three weeks this season after being struck on a foot by a shot in Vancouver's Oct. 3 season opener. He has recorded just three assists and is a minus-7 in 17 games.