The 36-year-old Swede was struck in the face by a puck and left the game for X-rays, but returned to finish the third period and overtime.
Alfredsson didn't require surgery, but will miss Thursday's home game against Vancouver.
"It's a fracture. It's just got to heal," said Alfredsson, who will likely wear a full face shield when he returns. "I need some time away from contact for a little bit.
"We'll probably look at it again next week and see where it is and go from there."
The injury comes at a bad time for a team that still has hopes of making the playoffs.
"Obviously, I'm not going to go out there and try to replace Alfie," Kelly said. "That's impossible."