Kane beat Kari Lehtonen high on the glove side with Chicago's second attempt in the shootout, and Crawford stopped Dallas tries from Tyler Seguin and newcomers Jason Spezza and Ales Hemsky after bailing out the sluggish Blackhawks for the first two periods.
''All night long he was great, so we can be happy getting out of here with a win,'' said Sharp, who had 10 points in five games - including six goals - against the Stars last season. ''But if it wasn't for Corey, it'd be a different situation.''
Crawford stopped 32 of 34 shots for the Blackhawks in their first meaningful game since losing Game 7 of the Western Conference finals to the Stanley Cup champion Los Angeles Kings last June. They won their seventh straight in Dallas.
''We have to call the cops. We stole two points tonight,'' Chicago coach Joel Quenneville said. ''Certainly call that a goalie win.''
The Stars opened on the same ice where last season ended in a first-round playoff series against Anaheim after they broke a five-year postseason drought.
Dallas had turned away several good scoring chances for the high-scoring duo of Kane and Jonathan Toews before Sharp sent a shot from the circle past Lehtonen 11:10 into the third.
The Stars forced overtime by killing Jordie Benn's delay-of-game penalty in the final 2 minutes of regulation. Vernon Fiddler was stopped by Crawford on a good short-handed chance early in the penalty, and the two got tangled up while play continued at the other end. The Blackhawks didn't get a great scoring chance in what amounted to a 5-on-3 opportunity.
''I do feel good,'' Stars coach Lindy Ruff said. ''We had questions on our defense, and our defense was real good. That's going to be a fun game to look back at.''
The Stars outshot Chicago 24-11 through two periods before the Blackhawks controlled the pace in the third period and overtime.
Chicago had only one shot in the extra period but had the puck in the Dallas end throughout. Kane just missed on a spinning attempt and later sent the puck through an empty crease after some nifty stick-handling.
''It definitely was a pretty slow start for us,'' Crawford said. ''We gained our timing as the game went on and gained our confidence and that speed and puck possession game as we went further into the game. You could see that.''
Spezza and Hemsky assisted on a tying goal by Trevor Daley in their Dallas debuts after playing part of last season together in Ottawa. Spezza was acquired in a trade just before Hemsky signed a free-agent deal with the Stars.
The Stars scored again for a 2-1 lead in the second when Cody Eakin slipped a shot from just above the crease past Crawford.
The 23-year-old Eakin got the first opening-night point of his career less than two weeks after ending a contract holdout by signing a two-year deal worth an average of $1.9 million per season.
''If we work that hard during a game, we're going to win a lot of them,'' Eakin said. ''The shootout didn't go our way but for the most part we skated well and worked really hard and played our systems well.''
NOTES: The Stars lost for the first time in three season openers against the Blackhawks. They had previously won at home in Dallas and when the franchise was still in Minnesota. ... Stars D Alex Goligoski played his 400th career game. ... The Blackhawks won their third straight season opener after going winless in the previous three. ... Daley's goal came during a 4-minute high-sticking penalty against Niklas Hjalmarsson, who caught Patrick Eaves in the face.