Corey Crawford settled down after allowing a weak goal in the opening minutes. Marian Hossa also scored, and the Blackhawks took the early lead in this first-round series.
Game 2 is Friday at the United Center.
The Blackhawks finally put this one away when Johnny Oduya chipped the puck off the boards to Viktor Stahlberg on the right side. Stahlberg then dished it to Bickell on the two-on-one rush in front for the winner.