Marian Hossa's goal, from a brilliant no-look pass by Patrick Kane, gave the Hawks their first 4-0 start in 40 years.
They had to come from 2-0 down to force overtime, though, Patrick Sharp with a goal and an assist after Loui Eriksson and Ray Whitney had put the hosts in control.
There was some good news on the night for the Stars, who confirmed the signing of All-Star centre Jamie Benn on a five-year contract. He will make his debut either against the St Louis Blues on Saturday or the Columbus Blue Jackets on Monday.
Two of Chicago's Central Division rivals met in St Louis, where the Blues beat the Nashville Predators 3-0 with Patrik Berglund contributing a goal and an assist. Columbus went down 4-0 at Colorado, Matt Duchene with two goals and an assist.
In the North-East Division, the Ottawa Senators rode out to a 3-0 start by beating the Florida Panthers 3-1.
Chris Phillips, Erik Karlsson and Jason Spezza scored the goals after Tomas Fleischmann gave Florida the lead.
Ottawa's divisional rivals the Buffalo Sabres fell to their first defeat, 6-3 to the Carolina Hurricanes. Eric Staal scored a hat-trick and brother Jordan had two assists, while Jeff Skinner scored twice.
The Washington Capitals, in contrast to Chicago, have made their worst start to a season in almost 20 years after a 4-1 defeat to the Montreal Canadiens left them at 0-3.
Montreal's Andrei Markov continued his excellent start to the season with a goal and an assist and now has four points in three games.
There were contrasting fortunes for New York's two Atlantic Division teams, with the Islanders beating the Toronto Maple Leafs 7-4 with two goals each from Matt Moulson and Michael Grabner while the Rangers went down 2-1 to the Philadelphia Flyers.
The San Jose Sharks scored four times in the third period to secure a third successive win, 5-3 against the Phoenix Coyotes, while the Edmonton Oilers beat the Los Angeles Kings 2-1 after overtime.