The Blackhawks began the third down by one and were up by two goals midway through the period 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.
Detroit carried a 2-1 lead into the third but Michal Handzus tied it in the opening minute of the final period. Bickell scored about 5 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 have the fans on their side Wednesday night in Game 7 against seventh-seeded Detroit.
"If I would've told Detroit and Michigan we would play in Chicago in Game 7, I think everybody would be excited about that," Red Wings coach Mike Babcock said. "I love Game 7s.
"We've got a chance to push them out of the playoffs. It should be a lot of fun."
Frolik became the first player in league history to score two goals on penalty shots in the playoffs. He also did it two years ago in the regular season against Vancouver.
"I was kind of surprised that I was the first one in history," he said. "It's a little bit special."
Chicago's Corey Crawford made 35 saves and Jimmy Howard stopped 24 shots for the Red Wings.
The Blackhawks sent the series back to Detroit with a 4-1 victory in Game 5 on Saturday night, and then jumped in front on Marian Hossa's goal in the first.
But 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 4, Detroit 3
Series tied 3-3
• San Jose at Los Angeles,7 p.m., NBCSN
Series tied 3-3 Wednesday
• Detroit at Chicago,6 p.m., NBCSN