The Blackhawks, who trailed the second round series 3-2, began the third period down by one and were up by two goals midway through after an offensive flurry.
They needed the cushion because Damien Brunner scored with 52 seconds left to pull Detroit within one.
The Red Wings pulled their goaltender, but they were unable to score with the extra skater.
Michal Handzus tied the game at 2 in the opening minute of the final period, and Bickell scored about five minutes later. Michael Frolik’s backhander on a penalty shot at the 9:43 mark put the Blackhawks ahead 4-2 and silenced the once-raucous crowd.
The Blackhawks will be at home Wednesday night in Game 7 against seventh-seeded Detroit.
Frolik became first player in league history to score two goals on penalty shots in the playoffs, according to STATS LLC.
Chicago’s Corey Crawford made 35 saves, and Jimmy Howard stopped 24 shots for the Red Wings.
The Blackhawks jumped in front on Marian Hossa’s goal in the first. Patrick Eaves tied the game later in the period and Joakim Andersson put the Red Wings up 2-1 with a long wrister 10:11 into the second period.
Chicago coach Joel Quenneville made the first tactical move by starting Jonathan Toews and taking him off the ice soon after the puck dropped to get him away from Henrik Zetterberg.
When the Blackhawks went ahead 1-0 on a power play, both captains were on the ice.
Toews won the first faceoff against Zetterberg after the penalty was called, and ended up with an assist on Hossa’s goalmouth scramble score 3:53 into the game.
Eaves, who revived his career during the lockout-delayed season after having a concussion, scored for the first time this postseason off a rebound 18:51 into the first.
Detroit put Chicago on the power play twice in the first 3:32 of the second <0x2014> once for having too many men on the ice <0x2014> and Howard had to make just three saves while the Red Wings were short-handed early in the second.
The Red Wings went ahead for the first time when Anderson had the puck, which was on edge above the left circle, and flicked a wrist shot that fluttered so much Crawford couldn’t handle it.
Chicago got to the front of the net in the third period and it paid off.
Handzus was basically by himself when he scored 51 seconds into the third against a helpless Howard. Bickell wouldn’t let Detroit defenseman Brendan Smith move him away from Howard before he scored the winning goal, leading to Howard taking Smith’s stick and throwing it into the corner.
Frolik got a penalty try when he had a breakaway after blocking Carlo Colaiacovo’s shot and was hacked by him from behind. He took advantage by scoring easily on Howard’s glove side.
Detroit’s goalie was irate, screaming at an official while the Blackhawks celebrated their comeback.