Michael Graber, Rick Nash and Oscar Lindberg also scored, and Henrik Lundqvist stopped 26 shots for the Rangers. New York has won three straight playoff games at Madison Square Garden after losing six straight on home ice, including Game 3 of the first round against Montreal.
Jean-Gabriel Pageau, who had four goals in Ottawa's 6-5 double-overtime win in Game 2, scored again for the Senators, and Craig Anderson finished with 26 saves.
Game 4 is Thursday night.
Looking to avoid falling into a 0-3 series hole and with the home crowd buzzing from the singing of the national anthems, the Rangers came out aggressive from the start. They outshot the Senators 15-5 in the first period while building a 2-0 lead.
Ottawa came out with more intensity in the second period, getting the first three shots of the period. The Senators had a 12-9 advantage on shots in the middle period but fell further behind on the scoreboard.
Lundqvist, who leads all goalies with 289 saves in the playoffs, made a nice stick save in the opening minute of the second on Pageau, and both goalies made some nice stops through the middle of the period to keep the score 2-0.
Nash then increased the Rangers' lead with his 15th career postseason goal — third this year — with 7:39 remaining in the middle period. After the Senators' Erik Karlsson and Ben Harpur collided near center ice, Derek Stepan brought the puck down the middle and passed it to Nash, who fired it past Anderson from the left circle.
Lindberg made it 4-0 with 1:43 left in the period. J.T. Miller fought off a defender along the left side, skated in toward the net and passed it to Lindberg, who fired it over the glove of a sliding Anderson from the right side.
Pageau then spoiled Lundqvist's shutout bid just 32 seconds later as he got a pass from Bobby Ryan from behind the net and scored his sixth of the postseason.
Zuccarello got the crowd fired up when he got New York on the scoreboard 5:31 into the game. Mika Zibanejad brought the puck up the right side, came around the back of the Ottawa net while fighting off a defender and sent a pass in front to Zuccarello, who fired it past Anderson. It was Zuccarello's 11th career playoff goal.
Grabner made it 2-0 with 6:36 left in the opening period. Zuccarello sent the puck along the boards on the left side behind the Ottawa net and Grabner got it, came out and put in a wraparound before Anderson, who had gone to his right, could recover.
Lundqvist had a nice glove save on Karlsson with about 3 1/2 minutes to go, and Anderson stopped Zibanejad in front with less than 2 minutes remaining.
The Rangers have won at least one game in 21 straight postseason series since being swept by New Jersey in the first round in 2006, setting a franchise record. The previous mark of 20 was set from the quarterfinals vs. Boston in 1970 to the quarterfinals vs. St. Louis in 1981.
Nashville 2, St. Louis 1
Pekka Rinne outdueled St. Louis' Jake Allen with 32 saves, and the Predators are a victory away from reaching the Western Conference final for the first time in franchise history.
Ryan Ellis scored at 5:09 of the third to break up a scoreless game. Neal then scored with 6:57 left to crank up the Nashville celebration as the Predators won their eighth straight playoff game at home.
Joel Edmundson scored with 3:49 left as St. Louis avoided the shutout. Blues coach Mike Yeo pulled Allen late, but they couldn't beat Rinne again. Allen finished with 23 saves as the Blues lost consecutive road games for the first time since Feb. 28 and March 3.
Game 5 is Friday night in St. Louis.
The Predators got the first goal of the game after a scuffle along the boards in front of the Nashville bench put Edmundson and forward Ryan Reaves in the box for roughing along with Predators forward Cody McLeod. The puck bounced around before Ellis scored on a wrister from the edge of the left circle under Allen's head as the goalie went down.
Ellis tied the franchise record with a point in his seventh straight playoff game.
Just after Ellis' goal, Predators defenseman P.K. Subban had his helmet knocked off by Edmundson along the side boards, sending him to the locker room. Subban returned later in the period after being called for embellishment along with Edmundson back in the box for roughing at 5:15 of the third.
Allen got help from the crossbar on a shot by Neal and Viktor Arvidsson hit a post. Neal finally scored his second this postseason intercepting the puck, spinning and hitting the top corner with a wrister.
That proved to be the game-winning goal after Edmundson scored with a shot from the left circle through two players in front that banked in past Rinne, giving the Blues a chance.
With a late start, the Predators tapped one of their original fans in country star Vince Gill to sing the anthem along with his daughters, and golfer Brandt Snedeker brought the Ryder Cup with him onto the ice before waving a flag to rev up the crowd.
Former Jets and Bills coach Rex Ryan also was in the stands for his second game this series wearing a Predators' sweater.
This was the first game of this series to go scoreless through the first 40 minutes thanks to dazzling saves by both goalies.
Rinne opened the game easily gloving a shot by Vladimir Sobotka and looked like the goalie who shut out Chicago twice as Nashville swept the Blackhawks in the first round.
He got what appeared to be a helping hand by Ellis in the final seconds of the second period with the Blues on the power play. Ellis blocked a shot from Jaden Schwartz, then Rinne made saves on Alex Pietrangelo and Schwartz in a flurry inside the crease.