Connolly was driven head-first into the corner boards by Flyers captain Mike Richards with 6:03 left in the second period. Connolly was chasing the puck into the corner and spent about two minutes prone on the ice before skating slowly to the dressing room.
Richards was assessed a two-minute boarding penalty on the play.
Buffalo was up 4-3 after two periods but the Flyers tied the game midway through the third and Flyers left wing Ville Leino scored the game-winner 4:43 into overtime.
Game 7 is Tuesday night.
Connolly has a history of injury problems, including several concussions. One of the concussions forced him to miss the entire 2003-04 season.
The Sabres provided no update on Connolly's status.
Information from The Associated Press contributed to this report.