The Sharks, who swept Dallas in a home-and-home series, have 92 points and remained eighth in the West, while the ninth-place Stars (89 points) are three points out with two games remaining.
San Jose hosts Los Angeles on Thursday in the start of a season-ending home-and-home series.
"We can't enjoy this for very long," San Jose coach Todd McLellan said. "We've got work to do tomorrow. Good teams keep their highs and lows at a very even keel and we still have work to do. We cannot let our guard down. That'll be the message again sent to the players tomorrow."
The Sharks drew first blood at 12:09 of the opening period when Winnik snapped a wrister past Lehtonen for his eighth goal of the season. The sequence began with a slapper by San Jose's Brent Burns from the point, an effort that was deflected wide of the Dallas net by Andrew Desjardins. Lehtonen dove to the ice in anticipation of Desjardins' redirect and Winnik saw the opportunity, taking a shot from the left circle which deflected off the Stars' goaltender and into the back of the net to make it 1-0.
Just as it looked like the Sharks were poised to head to the first intermission with the lead, Dallas pulled even when Morrow scored his 11th of the season at 17:47.
The Stars' captain flipped a backhand at Niemi from near the right post, that went off the goaltender's stick before going through the five-hole to tie the game.
"Giving up a goal like that, it hurts but I was able to keep playing and didn't think about that too much," Niemi said.
Late in the first, the Stars drew their first power play when Colin White was guilty of cross-checking at 18:38. Burns joined him in the box 1:02 later when he was whistled for hooking. However, Dallas was unable to convert turn those chances into goals.
Those would prove to be the Stars' only two power plays of the game as they finished 0-for-2.
"We just didn't execute there," Dallas coach Glen Gulutzan said. "The one there [late in the first], we didn't know the time left on the clock. We had a set play, got a shot off there, we just didn't execute and then going into the in-between periods, we had things set up, but we didn't execute."
Dallas took its first lead of the game at 11:10 of the second, when Goligoski blistered a long slapper from the point that beat a screened Niemi to his right. But San Jose answered just 32 seconds later when Galiardi beat Lehtonen top-shelf with a wrister from near the left post, making it 2-2.
Gulutzan called Galiardi's goal the turning point in the game.
"We made a mistake on coverage," he said. "It's always right after a goal you need a good shift. We just didn't get it. They got right back in the game. We had no chance to build on the momentum of that goal."
When Michael Ryder was whistled for tripping at 16:57, it put the visitors on the power play for the first time in the game. The Sharks wasted little time in converting -- 27 seconds to be exact -- as Couture ended a 10-game drought, redirecting a pass from Martin Havlat past Lehtonen for his 31st of the season. He hadn't scored since March 13 at Calgary.
"It feels good to score goals. It's nice to get one and help the team but we've got to move forward here," he said. "We've got two huge games coming up, looking forward to the game against L.A. [on Thursday]."
At 3:27 of the third period, Dallas defenseman Mark Fistric received a double minor for high sticking but the Stars' penalty kill rose to the challenge in what had to rank as one of their biggest kills of the season.
Niemi made his biggest save of the night at 14:15 of the third when he denied a Jamie Benn backhand on a breakaway. Less than a minute later, Thornton tallied his 17th goal of the season to help the Sharks pull away.
"Huge -- if he doesn't make that it's a different game," Thornton said of Niemi's save on Benn. "We've seen him do it in the past. We just kind of fell asleep. He's a world-class goalie and he makes that save, saved our bacon again tonight."
Lehtonen left the Dallas net with 1:57 remaining but even with the extra attacker, the Stars were unable to convert until Clowe hit the empty net with 1:04.
Dallas visits Nashville on Thursday before closing its season at home against St. Louis on Saturday. They have to win both games and hope for help.
"You have to move forward," Gulutzan said. "Now we need help and you just start preparing, getting ready for Nashville. There's no other way to look at it. There's no other way. It's just business as usual and go win the game."
1 - 0 SJS
1 - 1 Tie
2 - 1 DAL
2 - 2 Tie
3 - 2 SJS
4 - 2 SJS
5 - 2 SJS
Hi stick - double minor