Chris Kreider had a goal and an assist, Marc Stall added a score and Mika Zibanejad had a career-high four assists for the Rangers, who earned their first road win in regulation this season. Henrik Lundqvist made 34 saves.
''I'm not scoring the goals, I'm just trying to set them up,'' said Zibanejad, who centered a line with Kreider and Fast. ''We've been playing together before and it was really clicking back then. We just went back to how we played. We find each other pretty well out there. It's nice to see (Fast) come back and get a couple.''
''You can't lose six in a row,'' Bonino said. ''Not in this league, not in this division, not with the parity there is. Just gotta get better.''
The Rangers were trailing 3-2 entering the third, but Staal and Fast scored in just over a three-minute span early in the period, giving New York the lead.
Staal converted a wrist shot from the slot off of a 3-on-2 rush at 4:23, then Fast scored his second of the night off a slap shot from the slot to make it 4-3.
''There's a lot of different ways to win hockey games, and each game presents a different challenge,'' Rangers coach David Quinn said. ''This one presented a challenge of being down 3-2, getting two goals in the third and then having to kill a penalty at a key time and doing a great job on the 6-on-5.''
Lundqvist stopped a one-timer off the stick of Nashville captain Roman Josi with 23.2 remaining in the game with Rinne lifted for an extra attacker to preserve the win.
''It was definitely a battle, but it was a fun game to play,'' Lundqvist said. ''They're a good team. They're pretty balanced - good forwards, really good D, they move the puck well.''
Kreider scored the game's first goal at 2:42 of the opening period on a power play.
Above the right circle, Zibanejad faked a slap shot and then sent a pass to Kreider, who was just above the goal line to Rinne's right, where he was able to deflect the puck over the shoulder of the Finnish keeper.
Kreider's 20 goals lead the Rangers. He has scored in four consecutive games and five of his last six.
The goal came just seven seconds after Predators coach Peter Laviolette was assessed a bench minor penalty for unsportsmanlike conduct. Laviolette was unhappy with a hit former Predators player Cody McLeod delivered to defenseman P.K. Subban deep in the Nashville zone.
''It's a loss,'' Laviolette said. ''It's tough. They're mounting. We've got to put an end to it. We did enough good things through the course of the game. We got ourselves a lead going into the third. I think that's what makes it a tough one to swallow because typically we're good in that situation.''
Bonino tied the game at 1-all at 8:43 of the first.
From the left faceoff dot, Fiala sent a pass to Mattias Ekholm at the middle of the blue line. There, the Swedish defenseman sent a one-timer toward the Rangers net and Bonino was there to redirect the shot past Lundqvist.
New York made it 2-1 at 3:51 of the second. From behind the Nashville net, Zibanejad sent a pass to Fast, who was alone in the slot. He beat Rinne with a quick shot on the blocker side.
Fiala tied the game again at 4:44 of the second with an unassisted goal.
Driving down the right side on a 2-on-1, Fiala snapped a wrist shot low to Lundqvist's far side.
Bonino's second power-play goal of the game gave Nashville its first lead at 10:35 of the second after converting a rebound of his own shot at 10:35.
NOTES: Nashville C Kyle Turris missed the game with a lower-body injury. ...The Predators sold out all 51 home games they played at Bridgestone Arena in calendar year 2018. ... Rangers RW Kevin Hayes picked up one assist, extending his career-long point scoring streak to nine games. ... New York is 9-6-6 when scoring first this season.
Rangers: Visit St. Louis Blues on Monday.
Predators: Visit Washington Capitals on Monday.
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