Adam Fox scored a 5-on-3 power-play goal for the Rangers (2-2-0). Igor Shesterkin allowed four goals on 18 shots, and Jonathan Quick made nine saves after relieving him with 6:09 remaining in the second period and New York trailing 4-0.
It was Quick's debut with the Rangers. He signed a one-year contract July 1.
"All around it wasn't good enough to win a hockey game," New York captain Jacob Trouba said. "Even if somehow you find a way to win that game, that's not a game I felt we deserved to win at all. Definitely our worst game of the season so far to this point. So, address it. We don't want a repeat of this thing. We all saw it, we know what it feels like, address the issue and play better hockey. Learn from the experience."
Rangers coach Peter Laviolette said they allowed Nashville to have eight odd-man rushes and two breakaways.
"You're just not going to find success unless you button that up," Laviolette said. "That's something we've been talking about, something that needs to be addressed, something that is on topic every day when we go into a game to eliminate the high risk and to eliminate the standing at the offensive blue line. We had the problem against Columbus as well (5-3 loss on Saturday). I thought we gave up too much. But in Columbus we competed hard. We attacked the net hard. Tonight is different for me because there are a lot of components and aspects of the game that aren't good enough, that are missing."
Smith gave the Predators a 1-0 lead at 11:24 of the first period. Tyson Barrie's shot from the point hit Michael McCarron and got behind Shesterkin. Smith poked it into the net.
He made it 2-0 at 16:22, scoring on a breakaway after coming out of the penalty box. Rangers defenseman Erik Gustafsson couldn't handle K'Andre Miller's pass at the blue line, and the puck bounced into the neutral zone in front of the penalty boxes and right to Smith.
"Besides the goal part, just being able to contribute for the guys here and to finally bury one on a breakaway felt good," Smith said. "Any time you can put one in the back of the net, help the guys out and get a good start like that, it's good."
Ryan O'Reilly's power-play goal at 3:32 of the second period gave the Predators a 3-0 lead.
Forsberg made it 4-0 with his own power-play goal at 13:51, stick-handling through four Rangers and skating across the zone left-to-right before scoring on a slap shot over Shesterkin's blocker from the face-off circle.
"You could see he was pressing it a little bit earlier and his timing was probably off for him, and that happens," Predators coach Andrew Brunette said. "Him and (Josi) missed a big chunk of time at the end of last year and then they come in and it's hard to get your rhythm back, your cadence. Game by game they're getting better. They're elite players in this league and I expect them to continue improving."
Fox scored at 19:18 of the second for the 4-1 final.
Vincent Trocheck had a goal taken away after the Predators successfully challenged that he entered the zone offside with 8:05 remaining.
"The process wasn't there the whole night," Rangers forward Chris Kreider said. "Just got beat, flat out."
NOTES: O'Reilly has points in seven straight games he's played at Madison Square Garden since March 24, 2018 (three goals, five assists). … The Rangers have scored a power-play goal in each of their four games.