Shanahan scored his team-leading 25th goal in the third period for the Rangers, who have won two straight road games following the All-Star break. New York has beaten Boston six consecutive times, including twice this month.
"Maybe there's a desperation that players realize now the situation that we're in coming out of the break and just how important every game is," Shanahan said of the Rangers, who entered the night in ninth place in the Eastern Conference.
The Rangers were whistled for only four penalties, but registered two short-handed goals -- one by Cullen, and Shanahan's penalty-shot tally that made it 5-1. Michael Nylander closed the scoring with a power-play tally at 7:39.
"I really think that should be our identity and it can't just be in Philly, in Boston," Shanahan said. "There's a tendency when you're at home to want to open it up and put on a show for the fans and I think our fans just want to see that same battle level and they just want to see results and wins they want to see playoffs."
The Rangers face Toronto at home on Wednesday night.
The Bruins have lost three straight -- scoring just two goals -- and are 3-9-1 in their last 13 games. Patrice Bergeron got Boston even at 1 with a power-play goal in the first period.
"It's a devastating loss," said Boston coach Dave Lewis, who shook up his lines during the game and said more changes could come Tuesday at Buffalo. "We're going to pick ourselves back up and get ready for a hockey game tomorrow.
"We'll see what happens. The thing for me is don't quit, never surrender."
Cullen added two assists, and Ortmeyer had three as the Rangers' third line dominated the first two periods. Defenseman Michal Rozsival scored in the second, assisted by Prucha and Cullen.
Ortmeyer came into the game with three points (one goal) in nine games, following a 40-game absence because of a pulmonary embolism. He scored a penalty-shot goal in the Rangers' 3-1 win over Boston on Jan. 13.
Shanahan beat Hannu Toivonen with a backhander on the penalty shot for his third goal in 21 games. He then assisted on Nylander's goal 1:13 later.
"Any time we can take the pressure off those guys it makes their life easier," Ortmeyer said.
Prucha's second goal of the night, at 6:41 of the second period, chased Tim Thomas from the Boston goal. Toivonen, scheduled to start in Buffalo, yielded Cullen's 10th of the season at 10:00 on the first shot he faced.
Prucha, who has 12 goals this season, opened the scoring when his pass from behind the net hit Bergeron's left skate and went between Thomas's legs at 9:11 of the first. But Bergeron answered 1:44 later to tie it, beating Henrik Lundqvist off a scramble for his 16th goal and first in seven games.
Cullen's hard work down the boards got the puck to Prucha, whose pass found Rozsival alone in front. Rozsival misfired, but the puck found its way through Thomas' pads.
Lundqvist made 29 saves. ... Thomas allowed three goals on 15 shots and took the loss. Toivonen stopped 11 of 14. ... The Rangers scored a season-high seven goals in a 7-3 win at Phoenix on Oct. 28. Bergeron played in his 200th NHL game. Referee Tom Kowal was hit on the left leg by a long shot off the stick of Boston forward Glen Murray in the first. He went down but stayed in the game. Bruins trainer Don DelNegro worked his 1,000th NHL game. Aldo Reed, saxophonist for the Bob Seger Silver Bullet Band, performed the national anthem.
Copyright by STATS LLC and The Associated Press
Arena: TD Garden Location: Boston, Massachusetts
Referees: Tom Kowal, Dennis LaRue Linesmen: Steve Barton, Derek Nansen
Attendance: 13,367 (76.1% full)
Team Stat Comparison
Power Play Goals
Power Play Opportunities
Shots on Goal
Petr Prucha (11)
Assists: Jed Ortmeyer, Fedor Tyutin
Patrice Bergeron (16)
(Power Play) Assists: Glen Murray, Marco Sturm
Michal Rozsival (6)
Assists: Petr Prucha, Matt Cullen
Petr Prucha (12)
Assists: Jed Ortmeyer, Matt Cullen
Matt Cullen (10)
(Shorthanded) Assist: Jed Ortmeyer