The victory snapped a seven-game losing streak against the Bruins for the Devils (2-5-4).
"I mean, we were due for some good fortune," Devils coach Peter DeBoer said of his team's start this season, but he could as well have been referring to the recent games against Boston.
"We haven't had a lot of it through the first 10 or 11 games," he said. "I thought the guys continued to work hard, we dug ourselves a little hole there early and considering the circumstances, it would have been easy to pack it in and we didn't. We kept chipping away and like I said, we were due for some good luck."
Iginla and David Krejci each had two assists as Boston built a 3-1 lead.
With the Devils skating with a 6-on-3 advantage after Krug was given a double minor for high sticking Brunner, Patrice Bergeron was whistled for delay of game and they pulled Brodeur, Zidlicky unloaded a shot from the point that beat Tuukka Rask with 1:08 to play.
"It's a huge win," Devils' forward Patrick Elias said. "Obviously we didn't start well."
Green then scored the game-winner from the left circle on a 5-on-4 advantage at 19:15.
Rask made 28 saves.
New Jersey went 4 for 7 on the power play.
"Well, obviously the power play -- four power-play goals tonight," DeBoer said. "I don't think we've won in this rink in four or five years, so we were going to need something special tonight, whether it was a four-star performance from somebody or our power play to click. We got some of that."
Coming off a last-second win against San Jose on Thursday, Boston grabbed a 2-0 lead on its first two shots of the game against Brodeur.
Krug's power-play goal gave the Bruins a 1-0 lead at 7:52 of the opening period when he fired a slap shot from just above the right circle that ricocheted in off a Devils' player in front of the net.
Boston made it 2-0 just 1:27 later when Iginla sent a shot from the right wing that caromed off Brunner's skate and between Brodeur's pads. DeBoer then called time out. It turned out to be a small key in sparking his team.
Iginla didn't think the Bruins were overconfident after jumping on the Devils early.
"I think we're confident (when) we have the lead. We think we can hold it. It was just the second period that we weren't as good," Iginla said. "We weren't skating as well. We weren't first to the pucks like we were in the first period. You give them credit. They got better.
"But the third, with the lead by one goal up until the last four minutes, it was the way we want to play that. Being at home we'd like to put them away, but we weren't able to and they hung around and they got the power plays late and they took advantage of it."
It was Boston's third game in four days and coach Claude Julien thought it may have hurt his team, but didn't use it as an excuse.
"To me tonight we had one line going and too many mediocre guys," he said.
The Devils cut the deficit to 2-1 on Henrique's power-play goal when he swooped in and scored off the rebound off Eric Gelinas' shot at 11:24.
But Krejci sent Lucic in alone down the left wing and he fired a wrister past Brodeur with 56 seconds left in the first.
The Devils played a more spirited second period and had a handful of decent scoring chances before Brunner's goal sliced it to 3-2 with 27.1 seconds left in the period. He fired a backhander that slipped between Rask's right arm and the post, trickling across the line.
Boston had won 11 of the previous 12 games against New Jersey, with the lone loss in April 2011. ... Bruins D Zdeno Chara was whistled for two minor penalties. ... It was the first of three meetings between the teams this season, the only in Boston. ... Boston is off until Wednesday when it plays at Pittsburgh in a rematch of last season's Eastern Conference finals, swept by the Bruins. ... The Devils host Tampa Bay on Tuesday.