The Devils, known for tight defensive games, scored a combined 11 goals in victories over Philadelphia and Florida to open the season.
"They've been like this for what, 20 years now?" Tampa Bay goalie Evgeni Nabokov said. "That's New Jersey hockey. Anytime you play them that's what you're going to see, and everybody knows that. That's what makes their team dangerous."
Alex Killorn scored for the Lightning, who completed a season-opening four game homestand at 2-1-1. Nabokov stopped 23 shots in his Tampa Bay debut.
"It was just OK," Tampa Bay coach Jon Cooper said of the homestand. "Getting five out of eight points is decent. I truly believe you've got to be even better than that at home. But, five points is better than zero points."
With Lightning star Steven Stamkos serving a minor penalty, Cammalleri gave New Jersey a 2-1 lead from near the post at 11:19 of the third.
Killorn tied it 1-all by beating Schneider from in-close off a nifty pass from Tyler Johnson 8:59 into the third.
Zajac put the Devils up 1-0 with a rebound goal 7:38 into the second after Nabokov stopped the center's initial shot.
The Lightning had the first seven shots, including a wraparound by Nikita Kucherov that Schneider and forward Tuomo Ruutu combined to stop along the goal line. The no-goal call was upheld by a video review.
"Sometimes you need a break like that early on in a road game and we ran with it," Schneider said.
Nabokov made a post-to-post save on Michael Ryder's in-close shot late in the first and stopped Jaromir Jagr's third-period breakaway backhander.
"That's why people tell you good luck sometimes," Nabokov said of the Ryder save. "It hit a little bit of my stick. That was the luck."
Nabokov was signed as a free agent in July to backup Ben Bishop.
Ondrej Palat hit the post on an early second-period power play for Tampa Bay.
Schneider made a point-blank save on Valtteri Filppula during the opening minute of third.
"Obviously, Schneider was fantastic," Devils coach Peter DeBoer said. "We're getting contributions from a lot of different areas right now, and that's what it takes to win."
NOTES: Tampa Bay rookie F Jonathan Drouin (broken right thumb) resumed taking faceoffs in practice and expects to soon add contact drills. "We're just happy with the progress so far," Drouin said. "It's getting closer and closer now." Drouin still has some pain in the thumb. ... Jagr played in his 1,476th NHL game, which is eight away from tying Rod Brind'Amour for 18th place all-time. ... Lightning D Radko Gudas didn't play due to a lower-body injury. ... New Jersey F Patrik Elias had an assist and tied Dino Ciccarelli for 80th place all-time with 592.