The Blues' Colton Parayko unleashed a shot in the third period that hit Bishop on the left shoulder. Bishop went down, clutching his shoulder.
While Bishop was laying on the ice, the Blues' Jaden Schwartz scored to make it 3-1 with 13:23 left when he redirected Alexander Steen's shot. Steen had gathered the puck outside the right circle after Parayko's shot was deflected off Bishop.
Bishop was tended to by a trainer and initially stayed in the game, but he was subbed soon after with Anton Khudobin.
"He's fine. We just wanted to put [Khudobin] in; [Bishop] wasn't hurt. That's why we switched," Montgomery said after the game.
The Blues won 4-1 to force Game 7 on Tuesday.
Bishop, who came into the game with a .936 save percentage in the playoffs, stopped 16 of 20 shots. Khudobin had five saves in the final 12 minutes in his first action this postseason.
The Associated Press contributed to this report.