Defenseman Marc Staal also scored for the Rangers, who posted their second straight road win, when he came out of the penalty box and scored on a breakaway.
"A clear-cut breakaway from the blue line, definitely my first," Staal said, smiling. "I picked it up. I didn't know how close [Zdeno] Chara was to me, so I just tried to get away as fast as I could and go to the old trusty backhander. I don't get breakaways often, but I used it in junior every now and then, and in shootouts, and it's worked, so that's what I went to."
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Chara and Nathan Horton had goals for the Bruins, whose four-game winning streak was snapped.
In a spirited first period, highlighted by three fights, numerous breakouts and hard hits, the Rangers jumped ahead 2-0 on the goals by Anisimov and Frolov.
Anisimov's power-play goal made it 1-0 at 11:34. He batted the puck with a backhanded swipe over goalie Tuuka Rask's left shoulder after the puck bounced into the air. The goal held up to a video review when it was determined that Anisimov's stick wasn't above the crossbar when he struck the puck.
Bruins defenseman Dennis Seidenberg attempted to clear the puck in front of his net, but hit Frolov's lower body. The puck bounced over Rask to make it 2-0.
But it was Staal's goal that had the Rangers in a joyful mood.
The Rangers increased their lead to 3-1 on Staal's goal after Chara scored for Boston late in the first. Staal came out of the penalty box, picked up a loose puck near the Bruins' blue line and skated in on a breakaway, slipping a backhander past Rask 48 seconds into the second.
"When you have a forward [Brandon Dubinsky] and a D, you think the D's going to give the puck to the forward," Lundqvist said, breaking into a laugh. "But Staalsie just took off and he made a great move there."
Rask, making his first start since a season-opening 5-2 loss to Phoenix in the Czech Republic, made 27 saves.
"We felt good about his game," Boston coach Claude Julien said of his goalie's long layoff. "With a little luck, he would have had his first win tonight."
Bruins defenseman Andrew Ference thought a poor start hurt Boston as much as the lucky goals.
"They were tough goals, but we were outplayed in the first period," he said. "It's tough to take a whole period of hockey and talk about two goals."
Skating with their second two-man advantage of the first period, the Bruins cut the deficit to 2-1 when Chara fired a slap shot from the right point inside the left post with 5 seconds left.
Horton's goal cut it to 3-2 midway into the period.
The Bruins had few good chances in the third. They pulled Rask with 43 seconds left for an extra skater, but managed just one long shot.
Rangers LW Brandon Prust was hit in the face by a high stick from Gregory Campbell in the second period, drawing a double minor. Rangers coach John Tortorella said he was taken to the hospital to have an eye injury evaluated. No further information was available. ... Horton has at least one point in each of Boston's six games, matching his career high. ... Rangers RW Ryan Callahan earned his 100th career point with a second-period assist. ... Boston D Johnny Boychuk headed to the dressing room after getting slashed by Dubinsky in the first period. He came back for a few shifts later in the period, but missed the rest of the game because of a hand injury. Julien said the team will know more Sunday. ... Boston had killed 14 consecutive penalties until Anisimov's goal.
Copyright by STATS LLC and The Associated Press
Arena: TD Garden Location: Boston, Massachusetts
Referees: Stephane Auger, Chris Rooney Linesmen: Pierre Champoux, Derek Nansen
Attendance: 17,565 (100.0% full)
Team Stat Comparison
Power Play Goals
Power Play Opportunities
Shots on Goal
Artem Anisimov (3)
(Power Play) Assists: Brandon Dubinsky, Michal Rozsival
Alexander Frolov (2)
Assists: Michael Sauer, Erik Christensen
Zdeno Chara (2)
(Power Play) Assists: Mark Recchi, Nathan Horton
Marc Staal (1)
Assist: Ryan Callahan
Nathan Horton (5)
Assists: David Krejci, Dennis Seidenberg