Trailing 2-1 as the third period began, the Blackhawks drew even 51 seconds into the frame when Michal Handzus netted the equalizer off an assist from Niklas Hjalmersson.
Five minutes later, Bickell poked in the rebound of a Jonathan Toews centering pass in the crease for a 3-2 lead.
Michael Frolik added a crucial insurance goal on a penalty shot after a questionable call against the Red Wings, and Chicago went on to stun Detroit in a do-or-die effort after coming into the contest down 3-2 in their Western Conference semifinal series.
Marion Hossa added a goal and an assist and Corey Crawford turned aside 35-of-37 shots for the top-seeded Blackhawks, who will now host a winner-take-all Game 7 Wednesday at United Center.
Detroit's Damien Brunner netted his fifth post-season goal with 51.7 seconds left, cutting the deficit to one, but the Red Wings could get no closer.