Brad Marchand, Charlie McAvoy, Nick Ritchie and Matt Grzelcyk scored in regulation for the Bruins, who rallied with two goals in the final 9:38 of the third period. Jaroslav Halak stopped 31 shots.
''Sometimes it's not going to be pretty, sometimes you're going to make mistakes, but it shows the character that we have,'' Coyle said. ''Every point is crucial from here on out.''
Kyle Palmieri, Travis Zajac, Miles Wood and Michael McLeod each had a goal for the Devils, who are 4-0-2 against Boston this season. They beat the Bruins 1-0 on Sunday.
Mackenzie Blackwood, who stopped 40 shots for his first shutout of the season Sunday, made 44 saves. Ryan Murray added two assists.
The Devils were lined up for another win against Boston before McAvoy scored with 9:38 left in the third and Grzelcyk tied it with 4 minutes to go on a wrister from above the left circle.
''That's the staple of our team,'' Marchand said. ''When we're hard on pucks and drive, we're a hard team to play. We took over that game in the third period.''
Coyle and Pastrnak both shifted around Blackwood before tucking the puck into the net as Boston scored on its only two shootout chances.
The Devils had the puck in Boston's zone for nearly all of a power play late in OT after McAvoy was whistled slashing Damon Severson.
''That's the disappointing part,'' Devils coach Lindy Ruff said. ''It's frustrating, disappointing our power play hasn't been able to break through and win these games.''
New Jersey fell to 0-4 in shootouts.
''The shootout is obviously a high-skilled part of the game,'' Ruff said. ''Not only haven't we scored (enough) to win a shootout, but we haven't stopped a lot either.''
Leading 2-1, New Jersey had increased its advantage when Zajac tipped in Jesper Bratt's shot from the point 3:45 into the second period.
The Bruins cut the deficit in half on Marchand's power-play goal, but Palmieri stole a clearing pass from defenseman Jeremy Lauzon in the slot before firing a forehand by Halak just 1:18 later.
Things got testy after New Jersey went up 4-2 with scrums and tussles after whistles.
The Devils led 2-1 after one period, but it could have been more because they had a handful of other good scoring chances due to breakdowns by the Bruins' defense.
Wood gave the Devils a 1-0 edge 1:28 into the game when he took a pass from P.K. Subban, who stole a clearing attempt at the point, and broke in alone before slipping a wrister into the net.
Ritchie's goal tied it just over four minutes later. The winger fired a seemingly harmless shot from the right circle that Blackwood fumbled trying to catch before the puck dropped and slid into the net.
Marchand returned after missing the previous two games because he was in COVID protocol. He said it was ''a false positive'' during a morning video news conference.
Marchand, the team's leading scorer (13 goals, 23 assists), said: ''it was a bit odd, but a lot of teams are going through it.''
Defenseman Brandon Carlo played for the first time since suffering a concussion on a head hit by Washington's Tom Wilson on March 5, that got Wilson suspended for seven games by the league.
''The first few days were kind of tough,'' Carlo said in a morning media session. ''My vision was kind of blurry and messed up. I couldn't really see out of my right eye. After that cleared up, I knew it was just going to be a matter of time.''
NOTES: Boston center Sean Kuraly and winger Jake DeBrusk are both still in COVID-19 protocol. They've been out since the team returned to practice last Thursday after two games were postponed. ... Ritchie's goal was the first the Bruins have scored 5-on-5 against New Jersey this season. ... Marchand and Bratt each dropped the gloves and got tangled up behind the Devils' net in the second period but got whistled for roughing despite throwing a few punches.
Devils: Host the Capitals on Friday night in the first of two straight at home against them.
Bruins: Face the Penguins on Thursday night, the first of two consecutive meetings in Boston.
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