Alexis Lafrenière scored twice, and Igor Shesterkin made 20 saves for the Rangers (35-19-9), who have lost six of eight (2-5-1).
New York is third in the Metropolitan Division, seven points behind the New Jersey Devils for second and seven ahead of the New York Islanders, who are in fourth.
"We didn't create enough in the first two [periods], but I thought overall it was pretty good," New York coach Gerard Gallant said. "We had a chance going into the third, 2-1 down with an opportunity, but we didn't (score). Like I said, there was a lot of good, but there was some bad too."
The Rangers played with 16 skaters (11 forwards, five defensemen) due to injuries to defenseman Ryan Lindgren (upper body) and forward Tyler Motte (upper body). Defenseman K'Andre Miller was serving the final game of a three-game suspension for unsportsmanlike conduct.
"I think that was a step in the right direction," New York defenseman Braden Schneider said. "Over these next couple days, we can get rested up and get some more bodies back and get rolling because I thought tonight was a good effort."
Coyle gave the Bruins a 1-0 lead at 18:07 of the first period. Trent Frederic backhanded the puck behind the net to Bertuzzi, who passed to Coyle at the net front, where he slid the puck five-hole on Shesterkin.
"They got a pretty good team over there. They got some players, some deadly players, but we have some good players too," Coyle said. "These new additions (Bertuzzi, forward Garnet Hathaway, defenseman Dmitry Orlov) who have stepped up on the score sheet and other ways, the little details, and that's been huge for our lineup with our injuries. … That's what we need. We need that depth and everyone playing that way and picking up the slack."