The Bruins won the battle of wills, and the game, going away.
Goalie Tuukka Rask made 24 saves and Kevan Miller, Simon Gagne and Loui Eriksson also scored for Boston (15-12-1), which avoided its first pointless road trip of more than four games since 1963 after losses against the Anaheim Ducks, Los Angeles Kings and San Jose Sharks to start this western road swing.
Boston scored five goals in a game for the first time since Nov. 6 and scored on the power play for the first time since Nov. 13, snapping an 0-for-19 drought.
"We didn't want to go home having not won a game here," Boston coach Claude Julien said after the Bruins beat the Coyotes for the fifth straight time. "I thought we played well enough on this trip that somewhere along the way we should come away with a win. This was our last crack at it, and we found a way to get a win we desperately needed."
Joe Vitale and Mikkel Boedker scored for Arizona, which has lost seven straight at Gila River Arena (0-5-2), the longest losing streak on home ice in the original Winnipeg Jets'/Coyotes' 44-year history.
Devan Dubynk made 20 saves before he was pulled after the fourth Boston goal. Mike Smith played the final 24 minutes and made eight saves.
"It [stinks] to be in this. We all hate losing in here," Boedker said. "We all want to find a way out of this, but the only way we do is by sticking together."
The Coyotes (10-15-3) made a flurry of roster moves following a 4-0 loss against the Los Angeles Kings on Thursday, but lost for sixth time in the past seven games overall. They have been outscored 12-2 in the past three home games and are now 4-7-2 at home this season.
"Both teams came in desperate and we shot ourselves in the foot with early turnovers," Arizona coach Dave Tippett said. "It's a hard league and it's hard to win. This whole string has been tough to chew on."
The Bruins, who led 2-0 against the Sharks 2:35 into play on Thursday but went on to lose 7-4, scored in the first minute for the second straight game.
Arizona defenseman Oliver Ekman-Larsson's long cross-ice pass in the neutral zone was picked off by Hamilton. Hamilton hit Smith in the slot and he set up Marchand, who beat Dubnyk to the shot side with a wrist shot at 58 seconds to start his big night.
"We played really good right from the get-go and everyone was buying in and doing what we had to do," said Marchand, who has eight goals. "It's one of the better games we've played in a while and hopefully we keep building on it."
It became 2-0 when Dubnyk played the puck behind his net but his wobbly pass went right to Miller on the sideboards. Dubnyk stopped one quick shot attempt, but Miller ran down his rebound untouched and beat Dubnyk between the pads at 5:58. Miller's goal was his first in 54 games, dating back almost a full year (Dec. 8, 2013 against the Toronto Maple Leafs).
"We gave them the first two goals, just plain turnovers by us," Tippett said. "Both teams come in desperate to win and we shot ourselves in the foot. We have to take those plays out of our game.
"But we need our goalies to clean up a little more. It's been an issue all year except for a few games. If one of them wants to take the ball and run with it and win a bunch of games, I'm all good with that."
After going more than three full games without a goal on home ice, the Coyotes broke through on a play that started with a nice Dubnyk save in tight on Daniel Paille. Michael Stone started Shane Doan on the rush and his pass found Vitale in stride. Vitale's wrist shot from the right faceoff dot beat Rask to the top far corner at 13:11 and Arizona's first goal at home in 185 minutes, 34 seconds cut the Boston lead to 2-1.
The Bruins, who blew a two-goal lead for the 15th time in the past 155 games (140-9-6) at San Jose, didn't let this one get away. They buried the Coyotes with a three-goal second period.
It started at 8:34 when Dubynk stopped an innocent-looking shot by Paille. The rebound hit the shin pad of Gagne, who was standing right of the post and behind the goal line, and the ricochet squeezed between Dubnyk and the right post. The initial call of a kicking motion on the ice was overturned by video review and Boston's two-goal lead was restored.
"When it's 2-1, the message on the bench was we have to get the next one," Julien said "We either let them back in the game and start from scratch or we get a two-goal lead again. We haven't had many breaks lately so it was nice to get one."
Marchand made it 4-1 at 14:47, finishing off a 2-on-1 with a pass from Smith and chasing Dubnyk from the game.
"We all really wanted this win and it showed," Rask said. "It's been weird this year. We talk about [how we want to play] and then we show it for 20-30 minutes and then things just collapse within games. Today we focused on really battling and leaving everything out there."