Zajac scored the winning goal in a shootout, and Dowd notched a pair of goals in regulation, as the Devils broke a season-high, three-game losing streak with a 4-3 victory over the Florida Panthers on Saturday night.
The victory also kept New Jersey from slipping into a tie for first place in the Atlantic Division with Pittsburgh.
"No comments [were made about Pittsburgh]. Not from the coaching staff," Devils coach Claude Julien said. "I think everybody probably knew but nobody talked about it. We didn't [talk about Pittsburgh] on purpose.
"Our focus wasn't, 'Well, we've got to win the hockey game because ... 'We've just got to win because we want to win the hockey game no matter what the situation is."
Florida's Ville Peltonen and New Jersey's Zach Parise each scored in the first round of the shootout, but Nathan Horton failed on the Panthers' chance in the second round. Florida captain Olli Jokinen then fired wide on his club's final attempt.
"A couple of sloppy plays," Peltonen said. "And they got the momentum."
Dowd broke out of a 4 1/2-month scoring drought with a pair of goals. Both of Dowd's goals came with Florida up by one.
"It's great," Dowd said. "It felt real good, obviously. It's been awhile and it felt great. I think it might have been my first (two-goal game) to tell you the truth. It's a great feeling and then to win the game, I mean it's great, you know?"
Florida broke a 2-2 tie on a goal by Peltonen just 2:31 into the third period, but Dowd tied it when he beat Ed Belfour.
New Jersey's Martin Brodeur matched his career high with 43 wins, a total he reached three previous times.
The slumping Devils were 3-5-2 in their previous 10 games and needed to win to stay ahead of the Penguins, who beat Atlanta in Pittsburgh earlier Saturday.
Peltonen gave Florida a short-lived lead when he rebounded the puck into the empty left corner of the net after Brodeur made a pad save on Chris Gratton.
The Panthers also failed to hold a 2-0 advantage.
"In the second period, we were controlling the play, but a couple of mistakes and the puck's in our net," Panthers coach Jacques Martin said. "And then we came back and got the third goal to take the lead, and it's not like they were putting pressure on us, we just got caught on a change and they capitalized."
David Clarkson cut New Jersey's deficit to a goal with 6:54 left in the second when he swatted the puck past a sprawled Belfour after Juraj Kolnik was sent off for hooking. It was Clarkson's second goal in five games since being recalled from the minors.
"We all kept saying the same thing," Julien said. "Let's get the next one, the next goal and we're right back in it. That's what we needed to have, a positive atmosphere around the dressing room and the bench."
Dowd tied it 3:48 later on a rebound, beating Belfour on his glove-side. It was his first goal in 55 games. Brodeur then received the secondary assist on Dowd's third of the season.
"I'm happy I assisted on one of his goals, also, and it was a nice one. I stopped it clean behind the net," Brodeur said with a laugh. "Jimmy's been great for the hockey club and for team chemistry, being positive. So for him to chip in a little bit it's great.
"You can tell when guys appreciate a guy when he scores and everybody laughs and everybody has a big smile."
Florida went up 1-0 on Jokinen's 36th goal at 12:18 of the first. Jokinen sent the puck over Brodeur's left shoulder from just above the left circle. He has nine points (five goals, four assists) in four games.
Jozef Stumpel missed a chance to add to Florida's lead when his shot caromed off the crossbar just above where Jokinen had scored with 2 minutes left in the period during a Panthers power play.
Florida, which will almost certainly miss the playoffs, was 15-3-3 when leading after the first period.
"Sure [they were desperate," Weiss said of New Jersey. "That's what they're about. ... When we start being fancy and not getting the puck out of our zone that's when we get into trouble. We had two leads and one two-goal lead, and at this level those have to be pretty much automatic and especially in the position we're in.
"It's a tough way to lose."
Belfour extended his franchise-record for consecutive starts to 26. ... Florida is 13-4-3 at home since Dec. 25. ... With 151 goals, Jokinen is one behind Pavel Bure for second place on Florida's career list. ... Devils LW Patrik Elias was scratched for the fifth straight game due to a groin strain. ... Dowd hadn't scored since Nov. 7 against Carolina. ... Brodeur won 43 games during the 1997-98, 1999-2000, and 2005-06 seasons.