"[Josi] made a good play to get it to the middle," Neal said. "I managed to get it back on my stick there and just got it on net as quick as I could. It felt good to see it go in."
Nashville (36-12-6) is 22-3-1 at home and has won 13 of its past 14 games here.
"We're coming in with a good mindset every game," Neal said. "Putting pucks to the net, getting to the offensive zone, all the things that make us a great hockey team. Playing with speed, being physical. I think we've played some really good periods and we've laid off in some. I think our mentality is we're going to win every game. It's great for our team."
Tampa Bay (34-16-6) scored twice in the third period to tie the game and was able to use its speed to create chances. Nashville goaltender Pekka Rinne said it was one of the fastest games he's been a part of this season.
"Some of the Eastern [Conference] teams, the play a fast game," Rinne said. "They take some chances, they have a lot of skill in their lineup. I thought that [Tuesday] and the New York Rangers game (a 3-2 Nashville win Saturday), that was a really fast game. They have some speed in the East. It feels like they play a little bit more open game, kind of trade chances a little bit more than in the West. I thought that we did an excellent job of matching with that and didn't give up too much. We were able to walk away with two points. That's all that matters."
Seth Jones gave Nashville a 1-0 lead at 8:31 of the second period on a wrist shot from the faceoff circle. The goal was Jones' sixth of the season, third in his past five games.
"He's extremely capable," Predators coach Peter Laviolette said. "We were layered for the first half of the season with a healthy group [on defense]. There's just not enough minutes to go around. I tell you, when I was at the World Championships with Seth (in 2014), it was my real first look at him. To watch him play the amount of minutes that he did, all situational play against the most quality opponent, I left there with a tremendous amount of confidence in him as a player."
Taylor Beck gave the Predators a 2-0 lead at 12:47 of the second period on a rebound from the crease. Paul Gaustad took a shot from the point that hit the post, and Beck was able to score past Vasilevskiy.
Brett Connolly made it 2-1 at 1:20 of the third period and became the 10th Tampa Bay player with at least 10 goals. Valtteri Filppula took a shot that hit the post and went to Connolly, who was able to put it past Rinne.
"[Connollly] got us one early there, and I think that gave us a lot of life when we got that one early," Lightning coach Jon Cooper said. "From there, I think we had a good rhythm going. The TV timeouts came at a really good time. That was one of those things where we almost could have won in regulation, but we went to overtime and they made one more play than we did."
Nikita Kucherov tied the game 2-2 at 7:50 of the third period when he swatted the puck out of midair past Rinne. Kucherov took a pass from Ondrej Palat, tipped it in the air with his stick and scored his 21st goal of the season.
"Down 2-0 in arguably the hardest place to play, if you're going record-wise in the League, against the team with the most points in the League," Cooper said. "You're down 2-0 and you eke a point out of it, can't be displeased with that."
Nashville forward Viktor Stalberg had an assist on Jones' goal after being recalled from the Milwaukee Admirals of the American Hockey League on Tuesday. Stalberg was in the lineup in place of Colin Wilson, who missed his second consecutive game with an upper-body injury.
"[Stalberg] has played well every time he's got a chance," Neal said. "He plays with a lot of speed and he's got a great shot. He rang one off the post there early, and he gets in there. He played a good game tonight."
Tampa Bay forward Cedric Paquette was scratched after sustaining an upper-body injury at the morning skate.