Marian Hossa scored two goals and Chicago defeated Dallas 5-2.
Patrick Sharp assisted on the Blackhawks' first four goals.
The Stars failed to score on any of their six power plays.
Chicago's penalty-killing unit was ranked last in the NHL, but it was good enough to stop all of Dallas' power-play chances. The Stars are 0 for 24 power plays at home this season.
Hossa said the Blackhawks' penalty killing is "definitely getting better and better, and we're getting on the same page."
"Technically, we didn't get a power-play goal," Dallas coach Lindy Ruff said, "but (the Stars' first goal) came at the end of one and it was still 5-on-4.
"We had some good opportunities we didn't convert. We got some good looks that could have been a game-changer right there."
In contrast, Kane's goal was just 23 seconds into Chicago's only power play of the game, 18:15 into the second period. It was the Blackhawks' only shot with a man advantage.
"It was frustrating with all of the penalties early," Sharp said. "We were patient and we capitalized on the power play. We knew it was going to come sooner than later."
The Blackhawks used two short passes across the ice, from Sharp and Andrew Shaw to Kane, to set up that go-ahead goal at 18:15 of the second period.
"The changing point was on a turnover by myself," Eakin said. "We have to be a little sharper. I thought we played a good game, but we had a few too many turnovers."
Ruff said: "They took advantage of a big mistake. We had a really big error on killing a penalty and should have walked out of the second period probably at 2-2.
"In the end, we gave the game away, and that hurts."
The Blackhawks took advantage.
"Basically, it was one of those nights where special teams was the differential," Chicago coach Joel Quenneville said. "Our penalty kill led the charge. It was a great goal for the power play. I thought overall we played a real strong game discounting the penalties we took."
Before that, the Blackhawks twice rallied from one-goal deficits.
Hossa evened the score 1-1 in the first period with an assist from Toews across the goalmouth, after Eakin had scored for Dallas.
Hossa said: "As soon as (Toews) got in position, I tried to find open space for putting the puck in the net, and he found me beautifully. He did an excellent job behind the goal."
Eakin scored at the end of what was virtually a 4-minute power play because of consecutive minor penalties against Chicago.
Hjalmarsson had barely returned to the ice from serving a tripping penalty when Trevor Daley took a backhand shot.
Eakin scored at 6:10, 2 seconds after Hjalmarsson's penalty had expired. He had gone into the penalty box just 1 second after teammate Brandon Saad's high sticking penalty ended.
Chicago's late first-period rally pulled the Blackhawks within one shot on goal of the Stars at 11-10.
The Blackhawks continued to frustrate the Stars' power play, killing off two more in the period.
Chicago tied the score 2-2 at 10:19. Lehtonen stopped a short-range shot by Toews, but Toews maneuvered around defenseman Brenden Dillon to knock in his own rebound. Lehtonen finished with 26 saves.
Sharp credited his teammates for his four assists.
"One of those nights where you try to make the right play, and my linemates were putting the puck in the net."
Dallas' goalie stopped a breakaway by Kane 23 seconds before the period ended.
The final two goals came 34 seconds apart in the third period. Hossa scored on a breakaway, and Hjalmarsson followed with a shot the length of the ice for his empty-net goal at 18:53.
The Stars finished with a 33-31 advantage in shots. . Dallas is the only NHL team that hasn't scored on a power play at home this season. ... The Stars had been on a three-game winning streak. Their previous seven games had been decided by one goal. ... Chicago center Michal Handzus was inactive for the seventh consecutive game because of an upper-body injury.