Neal beat Matt Murray at the side of the net with 22.6 seconds remaining in the period, banking the puck off the Pittsburgh goaltender's left arm.
The turning point for Rinne came in the second period.
With the Nashville fans behind him in Game 4, however, expect another big showing from Rinne.
The first Stanley Cup Final not only in the franchise's history, but also the first major league championship game ever played in Tennessee with Game 3 against the Pittsburgh Penguins on Saturday night. Then he gave up five goals in the span of 15 shots. "We have to be better on the power play". Since the National Hockey League took control of the Stanley Cup in 1926, no rookie has ever led the playoff field in goal-scoring, as Guentzel now does. The power play has been extremely poor for the Penguins for several games now.
Neal then got in on the scoring himself, making it 3-1 in favor of Nashville with 23 seconds left in the middle frame.
Rinne rewarded Laviolette's faith in him by playing his best game of the Final. In fact, teams that win the first two games of a Cup Final at home are 0-8 in road Game 3s since the NHL's 2004-05 lockout.
Meanwhile, Nashville's power play struck again for a key goal early in the second, which kick-started a three-goal run and gave the Predators' belief.
Mattias Ekholm concluded the scoring at 13:10 of the third with Nashville's second power-play goal of the game.
Notes: Smith is the 18th different Nashville player to score a goal this postseason, and Ekholm became the 19th.More news: India, France vow to tackle terrorism jointly
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The second period of this game was the blueprint for the Predators success in the playoffs.
First time Crosby and Malkin had no shots on goal combined in any playoff game and who knows how long it has been. I thought my breath smelled great.
Fans were a bit skittish after Pittsburgh was the first to score and we wondered if Pekke in fact WAS back on his game.
Fans chanted "Pek-ka! Pek-ka!" in support of Rinne. The Pens' only option is to put game 3 behind them and look ahead.
One local Uber driver had even been driving people around wearing a P.K. Subban jersey, happy to discuss the local team that is finally having its moment. 3, Nashville, Gaudreau 2 (Watson, Josi), 6:33.
"His game tonight was rock solid", Laviolette said.
By the time Jackson took to the stage in the late afternoon, the iconic strip lined with countless honky-tonk bars, closed off to traffic for the special free concert, was packed with Predators fans nestled shoulder to shoulder in a sprawling, seemingly endless sea of gold.
Based on the morning skate at Bridgestone Arena, P-A Parenteau and Harry Zolnierczyk looked to be replacing Cody McLeod and Vern Fiddler.
A lot of that had to do with the Predators, who claimed in the three days between Games 2 and 3 that all they had to do was clean up some things in their own end and be a little sharper and their fortunes would change. Fans packed shoulder to shoulder almost an hour before Jackson's concert with the street then being used for an outdoor viewing party. From there, it was mostly all Nashville, and it's probably fair to say the Predators have carried the play for all but three periods in this series.
The defending champion Penguins lead the best-of-seven series 2-0.