Corey Crawford settled down after allowing a weak goal in the opening minutes. Marian Hossa also scored, and the Blackhawks took the early lead in this first-round series.
Game 2 is Friday at the United Center.
The Blackhawks finally put this one away when Johnny Oduya chipped the puck off the boards up to Viktor Stalberg on the right side. Stalberg then dished it to Bickell on the two-on-one rush in front for the winner at 16:35.
Big things are expected in Chicago after a spectacular regular season that included a record start and the team's first Presidents Trophy since 1991.
The Blackhawks are eyeing a run to the Stanley Cup for the second time in four years. They have been eliminated in the first round the past two seasons after beating Philadelphia for the championship in 2010, and they realize that for all they accomplished thus far in 2013, they'll ultimately be judged by what happens in the playoffs.
They seemingly caught a break before the game when Minnesota goalie Niklas Backstrom was scratched because of a leg injury suffered while reaching for a puck in the pregame warm-ups.
Josh Harding replaced him and more than held his own after being limited to just five games following a multiple sclerosis diagnosis last summer. Harding made 35 saves.
The Wild, trying to match what the Los Angeles Kings did a year ago and win the Stanley Cup as an eight seed, took the lead just under five minutes into the game when Cal Clutterbuck fooled Crawford with a soft shot from the left circle.
The goal on Minnesota's first shot brought back some bad memories for Blackhawks fans who remember him allowing several soft goals in the playoffs against Phoenix a year ago.
Hossa tied it just over two minutes into the second period, streaking down the left side, getting behind Minnesota's Jonas Brodin and firing the puck between Harding's pads after taking a feed from Patrick Kane.
Harding made a big save with his right pad from point-blank range on Kane after he beat Marco Scandella in the opening minutes of the third period to preserve the 1-1 tie.
Chicago appeared to take the lead midway through the third. Minnesota's Jared Spurgeon fell on a loose puck just outside the crease, and the Blackhawks kept swatting at it. Jonathan Toews knocked it into the net, but the whistle had already blown.
Blackhawks fans responded with some nasty chants, but they were letting out big sighs of relief two minutes later when a wraparound shot by Minnesota's Jason Zucker slid across the crease.
After five of their six playoff games against Phoenix last year went to overtime, the Blackhawks found themselves in a familiar spot.
They almost won it in OT when Hossa skated toward the net and drew Harding to the ice, but the goalie barely deflected his backhanded shot. Crawford then had a big blocker save on Zach Parise.
Notes: Kane is sporting a playoff mullet again. "I think it's fun," he said. "It's good for a good laugh or two. It's something that's been a little bit of a tradition." The youthful looking Kane gained national attention when he grew the mullet for the championship run in 2010 and he went with it again the following year. He went with a beard, instead, last year. But on Tuesday, he was all business up front and party in the back. ... Besides Emery, the Blackhawks were without Dave Bolland (lower body injury).
MORE HBJ HEADLINES
New details for CarMax in Lynnwood, but no opening date Providence partners with Walgreens on in-store clinics Briefs: Great Clips seeking stories of busy people Stores must stop drone sales at New Jersey airport Audi challenges Teslaís Model X with luxury electric SUV Costco sued over claims shrimp harvested with slave labor
Our new comment system is not supported in IE 7. Please upgrade your browser here.