The Rangers have 31 points in Gallant's first 20. Phil Esposito held the previous mark with 30 in 1986-87.
"I would say it's a good feeling," Ryan Strome said of New York's 13-4-3 start. "But I also think that it's a group that's not satisfied. We're starting to feel like a winning team. Two-two in Boston in the past, we're young, we're a little fragile and I don't know what could happen."
Strome, Dryden Hunt, Alexis Lafreniere and Jacob Trouba also scored for the Rangers, who won their third straight game and seventh of eight.
Igor Shesterkin made 29 saves and Julien Gauthier set up two goals. Trouba's was an empty-netter.
Craig Smith and Patrice Bergeron each had a goal for Boston, which has lost consecutive home games after opening 6-1. Jeremy Swayman stopped 31 shots.
"They like to funnel pucks from all angles," Swayman said of giving up some rebounds. "They get good traffic in front. You've got to do whatever you can to prevent those rebounds."
Panarin batted home a pass from Gauthier from the edge of the crease for the Rangers' first lead of the game. Lafreniere scored with 3:38 to play on Gauthier's wing-to-wing pass during a 2-on-1 break.
The Bruins had taken a 2-1 lead in the second when Bergeron, just outside the crease, redirected Matt Grzelcyk's perfectly centered touch pass from the left circle into the net.
"It's early in the year," Boston forward Brad Marchand said. "We have a lot of new faces in our group. It's a different year for us, a lot different than we've had in a long time where we have to build something."
But New York tied it when Hunt scored off the rebound of Ryan Lindgren's shot from the point.
"I think for me it's just sticking to my game and keep doing what I'm doing," Hunt said of his time on a line with Panarin and Strome. "Obviously it's two special players."
Smith had pushed Boston in front 14:46 into the game when he stole an attempted clearing pass from defenseman Trouba and slipped a wrister past Shesterkin.
Shesterkin kept it a one-goal deficit with a nice left-pad stop on Taylor Hall's clean breakaway with about 90 seconds left in the period before Strome tied it 1-1 with 6 seconds remaining. Strome, positioned in the slot, collected Panarin's pass from the left circle and slipped a shot past Swayman's glove.
Boston dominated much of the play in the opening period, outshooting the Rangers, 17-5.
"I think it's everyone buying in," Strome said. "I think everyone realizes not every shift is going to be an offensive shift. ... I think we've done a good job of sticking to our system and trusting each other."
WHAT A STOP
Seconds after the Rangers tied it at 2, Swayman came diving across the crease to make a fully out-stretched stick save on Mika Zibanejad, who had a wide-open net.
"I was just trying to be in the right place at the right time," Swayman said. "It was definitely a lucky save, but I'm happy with that."
Boston defenseman Charlie McAvoy carried on a Bruins tradition on Thanksgiving morning, donating and delivering hundreds of pies to local organizations. He also purchased 300 winter hats to be distributed among shelters.
Rangers F Chris Kreider had a goal overturned on video review. He elevated the puck onto the blade of his stick and flipped it off the post with it traveling behind Swayman's back across the crease.
McAvoy was in the lineup after leaving the last game Wednesday in Buffalo early to get some stitches above his eye. He was checked by Zemgus Girgensons and went facefirst into the boards. ... The Rangers didn't get their first shot on goal until nearly halfway into the opening period despite having a power play, too. ... Boston D Brandon Carlo turned 26 on Friday. ... New York played in the NHL Thanksgiving Day showdown for the seventh time in the last nine seasons. The Bruins traditionally have a Black Friday matinee.
Rangers: host the Islanders on Sunday night in the first of a four consecutive games at home.
Bruins: host Vancouver on Sunday night in the second of three straight home games.