The Blackhawks got the game-winner when Viktor Stalberg blocked a clearing attempt by Coyotes defenseman Adrian Aucoin along the left boards.
The puck caromed off his skate directly to Bickel in the slot, who blasted it past Phoenix goalie Mike Smith and allowed Chicago to even its Western Conference quarterfinal series at 1-1.
Brandon Bolig scored his first NHL goal and Patrick Sharp also found the net for the Blackhawks, who rebounded from Thursday's 3-2 overtime loss in Phoenix.
Corey Crawford nailed down the victory with 30 saves.
Smith finished by stopping 46-of-50 Chicago shots.