On the winning play, Johnny Oduya sent a long pass along the right boards to Viktor Stallberg, who skated into the circle, drawing the defense.
Instead of shooting, he found Bickell alone in the slot, fed him the puck and the Blackhawks left wing cleanly beat Wild goaltender Josh Harding for the game-winner.
Marian Hossa added a power-play tally for top-seeded Chicago, which finished with the best record in the NHL and took the first game of its Western Conference quarterfinal series.
Corey Crawford earned the victory with 26 saves.
Cal Clutterback scored and Harding ended with 35 stops in an emergency start after Niklas Backstrom was hurt during warm-ups for the eighth-seeded Wild.