Steen beat Corey Crawford off a pair of short passes from Steve Ott and David Backes to cap the longest overtime game in franchise history.
"It's big. We want to start this series off on a good note," Steen said. "Especially at home in front of our fans."
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Ryan Miller blanked the Blackhawks after Patrick Kane scored on a breakaway to put Chicago up 3-2 late in the first period and Jaden Schwartz tied it with 1:45 to go in regulation.
Both teams face a short turnaround with Game 2 Saturday afternoon.
The Blues' previous longest overtime game was a 4-3 loss at Detroit in 1984 that extended 37 minutes, 7 seven seconds. The home record for a playoff overtime game was 33:49 extra time in a 5-4 win over Chicago on April 20, 1989.
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St. Louis had to kill off delay-of-game penalties for shooting the puck into the stands in the first two overtimes. The Blackhawks killed a holding penalty in the second overtime.
Jonathan Toews, like Kane back from a lengthy injury absence, had two assists for Chicago.
Schwartz, Vladimir Tarasenko and Adam Cracknell all scored their first playoff goal for the Blues.
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The Blackhawks kept the Blues penned in their own zone for 2:10 during the first overtime, and Maxim Lapierre made the save of the session getting his body on a drive by Kris Versteeg with less than two minutes remaining.
Tarasenko was among the best players coming off a 15-game absence because of a broken thumb.
Cracknell tapped in a rebound early in the first to end a scoring drought of 148 minutes and 39 seconds for a team that got shut out the last two games of the regular season.
Kane scored his 30th career playoff goal after catching the defense napping at the end of a St. Louis power play, beating Miller off a long lead pass from Toews for a 3-2 lead at 18:24 of the first.
Miller allowed three goals on just seven shots in the first with defensemen Johnny Oduya and Brent Seabrook also scoring for Chicago.
St. Louis' top line of David Backes, Steen and Ott was a combined minus-5 in the first. Ott was a bit of surprise, considering he was minus-7 during the Blues' six-game losing streak to end the regular season.
Crawford faced just three shots in the second, but needed big saves to thwart Tarasenko and Ott. He made glove saves on drives by Steen and Tarasenko not long before Schwartz got the equalizer.