Parise, who will be the Devils' lone representative at Sunday's All-Star Game in Montreal, notched his team-leading 28th of the season and tied Buffalo's Thomas Vanek for third on the league list. He now has six goals in his last seven games.
Elias added an empty-netter with 54.4 seconds remaining to reach the 20-goal plateau for the third straight season, and eighth time in his last nine campaigns. His two-point night moved him within 19 of John MacLean's franchise record.
Brendan Shanahan made his Devils debut in front of the home crowd, but was held without a point.
Clemmensen's fourth straight victory put him one away from his first career 20-win season. He has allowed four goals on 95 shots over his last three starts (.958).
Clemmensen has handled most of the work since Brodeur went down with a left elbow injury on Nov. 1, but Devils President, CEO and General Manager Lou Lamoriello announced during the first intermission of Wednesday's game that Brodeur has begun skating.
"Marty went to the doctor today and got the OK to skate," Lamoriello said. "He did skate today, but I wouldn't rush that by saying he's skating. It's part of the physical therapy that he'll be going through. It'll be a day-to-day situation as far as skating goes."
Brodeur had surgery on Nov. 6 and has missed 37 games. Lamoriello said Brodeur's recovery is progressing on schedule.
"[Marty has been] right on time, right to the day as far as what they said would transpire," Lamoriello said. "He's right on schedule as to what they thought. There has been no regression whatsoever."
The Devils took a 2-0 lead on All-Star starting goaltender Carey Price and led 3-1 after 40 minutes. Parise staked the Devils to a 4-1 lead in the third before Montreal closed back within two.
"We played very solid here tonight," said head coach Brent Sutter. "We played well. We were able to score at right times. We did a lot of good things tonight."
Martin scored on a breakaway just seven seconds after his first-period cross-checking penalty had expired. Elias flipped a perfect pass up the middle of the ice, and Martin skated in alone to snap his fourth of the season under Price at 11:52.
Zajac put the Devils ahead by a pair in the middle frame, when his follow up of Parise's rebound caught the left post behind Price. Price swept the puck over the line as he reached back with his catching glove at 6:18.
Zajac eclipsed his 14-goal output of a year ago with his 15th of the season. The third-year center has four tallies in his last five games.
Josh Gorges cut Montreal's deficit in half by lifting Saku Koivu's rebound over Clemmensen at 13:56 of the second. The Devils' goaltender made a terrific stop on Koivu's shot from the left circle, but couldn't recover to deny Gorges of his second of the season.
Oduya's blast from the top of the left circle with 1:40 left in the second was his fourth of the season and put the Devils in control. The club has yet to lose in regulation when leading after 40 minutes (21-0).
Parise was trying to pass across the slot to Zajac when his feed deflected off the stick of Sergei Kostitsyn and through Price 2:05 into the final frame.
A Matt D'Agostini slapshot from the top of the left circle through Max Pacioretty's screen at 7:46 was only the second goal allowed by Devils' penalty killers in their last 27 chances.
Elias nailed the empty-netter to clinch it. Colin White's assist on the goal was his 100th career point.
NJD NOTES John Madden recorded his second straight two-assist effort and extended his points streak to five games (1g-4a)... The Devils improved to 3-0-0 this season against Montreal... After scoring in five of its last six games, New Jersey's power play went 0-for-2.
1 - 0 NJD
2 - 0 NJD
2 - 1 NJD
3 - 1 NJD
4 - 1 NJD
4 - 2 NJD
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5 - 2 NJD