Nashville defenseman Mattias Ekholm slashed Eakin's stick out of his hands on a breakaway, setting up the game-winning penalty shot.
"I was trying to catch my breath first, and then I was trying to figure out what the heck I'm going to do," Eakin said. "They actually came out, a couple guys, and I don't know if they were buying their goalie some time or what, but it was giving me an extra breath. I went to the old faithful and shot it low blocker."
Dallas (38-31-10) remained alive in the Stanley Cup Playoff race with its first win against Nashville this season. The Stars (86 points) trail the Los Angeles Kings and Winnipeg Jets by six points in the race for the second wild card into the playoffs from the Western Conference; they have played one more game than the Jets and Kings, who hold the tiebreaker with more regulation/overtime wins (37-33).
"It's desperation," Stars forward Travis Moen said. "We needed to get that win, and I thought the guys came back and battled hard in overtime and did some great things to help us get that penalty shot. Just take it one game at a time."
Nashville (47-22-10) has lost its past three games (0-1-2), all at home, and leads the St. Louis Blues by one point for first place in the Central Division.
"We're just going game by game here, that's all," Predators coach Peter Laviolette said. "Worry about tonight, we'll straighten it out, head on the road and worry about that one."
Predators forward James Neal tied the game 3-3 with 1:55 remaining in the third period with goalie Pekka Rinne pulled for an extra attacker, one-timing Mike Ribeiro's pass by Stars goaltender Jhonas Enroth for his 23rd goal. The assist was Ribeiro's 500th in the NHL.
Neal, defenseman Shea Weber and forward Matt Cullen returned to the Predators lineup. Weber missed three games because of a lower-body injury, Neal nine games because of an upper-body injury, and Cullen five games because of a lower-body injury.
"I thought James looked good," Laviolette said. "He scored a big goal for us late, was able to contribute offensively. I thought the line was productive."
Nashville defenseman Ryan Ellis made it 3-2 at 10:46 of the third on a wrist shot from the slot that beat Enroth on the glove side for his eighth goal.
"We had two or three opportunities to put a dagger in them in the third," Stars coach Lindy Ruff said. "I thought (Jason) Spezza could have buried them. A tough change cost us a goal and gave them a little bit of energy."
Colton Sceviour gave Dallas a 3-1 lead at 7:28 of the third with his eighth goal. He crashed the net and shot the puck into a scrum in front, and the puck trickled past the Nashville defenders and Rinne.
"We're still alive," Eakin said. "We're just continuing one game at a time, one period at a time. Just give it everything we've got and see what happens in the end."
The Predators took a 1-0 lead at 9:59 of the first period on a Forsberg wrist shot that got past Enroth. Ekholm took the original shot that bounced off of a body in front and went to Forsberg, who put it in an open net for his 24th goal.
It was Nashville's only lead of the game. The Stars scored the next three before the Predators rallied in the second half of the third period to tie it.
"There was a lot of positive things to take from tonight, but at the end of the night, you've got to win the game," Laviolette said. "I think we could've played a cleaner game. There's things we could have done to keep the game a little cleaner and manage the puck a little bit better, but we've got to win a hockey game and we didn't do that."
Moen tied the game 1-1 at 4:09 of the second period with a wrist shot that got past Rinne.
"Just went to the net," Moen said. "[Sceviour] made a great play. He threw it on net, and it just happened to kind of bounce up to me, and I just shoveled it in. It was one of those plays where I was trying to drive the net, and [Sceviour] made a heads-up play and got the puck to the net. It was just a lucky bounce, and it came right to me."