Lehtonen made 26 saves in his second game since returning from a head injury sustained on March 11 in Dallas' previous victory.
The most important stop came after Eakin's tying goal at 15:07 of the second. Ottawa's Erik Condra took a shot from in front that Lehtonen stopped, but he and Condra ended up in the net. The play was ruled no goal.
"I was able to make the save," Lehtonen said. "The puck was in front of my pads. He got pushed from behind, and he pushed me into the net.
"It could have gone the other way too. It was good to see (the referee) was on top of the play, so I wasn't too worried about it."
Lehtonen lost at Philadelphia on Thursday.
"I felt good in Philadelphia. The result wasn't very good," Lehtonen said. "(Saturday) I got a couple of good bounces and a couple of posts, and the end result was good."
Senators coach Paul MacLean said that teams create their own good fortune.
"If you don't feel you have any puck luck, then you're not working hard enough on a day-to-day basis," he said. "You shouldn't expect a break or something to go your way unless you're out there doing the work."
"It feels like it's been a long time (10 games) for me," Eakin said. "It was nice to chip in and relieve that top line. We get momentum when we play four lines and everybody is creating opportunities and putting pucks on the net."
Jamie Benn, part of that top line, reached 30 goals for the first time in his five NHL seasons.
He took a pass from Tyler Seguin at the top of the left circle and fired a shot into the upper left corner of the net.
Lehner made 39 saves against Dallas, which had a 42-27 shots advantage.
The Senators goalie had allowed 19 goals in losing the previous four games.
"I thought he came back, and I think that's really a good part of the learning experience for him after things didn't go his way," MacLean said. "I thought he came out and he really battled."
Dallas outshot Ottawa 17-7 in the first period, but the Senators scored the only goal.
Colin Greening made a pass from the top of the right circle to the high slot to Karlsson, who fired a shot past Lehtonen at 3:27.
"We got the lead, but couldn't really follow through until the end there, and they had a couple of nice goals to finish it off," Karlsson said. "Unfortunately, just couldn't put the puck in the net."
Dallas failed to score on three power plays in the second but the Stars broke through during 4-on-4 play with 4:53 left in the period.
Lehner stopped Valeri Nichushkin's attempt as he skated across the crease, but Eakin poked the rebound out from under the goalie to tie it.
C Shawn Horcoff returned for Dallas after missing four games because of a broken finger. LW Erik Cole's streak of 269 consecutive games played ended because of an upper-body injury. He is day to day. ... Despite their recent struggles, the Stars are 6-1-1 in their last eight home games. They are 5-1 in games in which defenseman Daley has scored. ... Seguin extended his point streak to eight games (six goals, nine assists). ... Ottawa defenseman Karlsson has points in six consecutive games (two goals, six assists). ... In the teams' previous four games, they allowed 39 goals and 262 shots combined.
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DALLAS, TX - MARCH 22: Andrew Hammond #30 of the Ottawa Senators tends goal during warm ups against the Dallas...