Yanni Gourde and Ryan McDonagh scored for the Lightning (0-2-0), who lost two straight to start the season for the first time since 2008. Andrei Vasilevskiy made 24 saves.
"I actually liked the way we were playing [in the first period] and then we gave one up and it seemed like we kind of lost our confidence and we played more like to lose the game than to win the game," Lightning coach Jon Cooper said. "This is a tough league to go down 3-0."
Meier gave the Devils a 1-0 lead at 8:30 of the first period when he stuffed in a rebound from close range off his initial shot that was blocked.
Brown extended New Jersey's lead to 2-0 at 10:17, when he chipped in a goal from the low slot off a centering pass from Gritsyuk.
It was Gritsyuk’s first NHL point in his second NHL game.
"He plays with a lot of speed, and I think I play with a lot of speed," Brown said of Gritsyuk. "I think we can turn the corner pretty quick. He's a really good player, so it's going to be fun."
The Devils went up 3-0 at 13:48 with Hischier's first goal of the season, a wraparound shot that deflected off of the skate of Lightning defenseman Max Crozier.
"In the first period they came out and tried to get under our skin, but we stayed calm and handled it the right way," Meier said. "And we came out of it with a 3-0 lead. We put pucks on net and won our battles. I thought we did a good job in the first period winning a lot of battles that we didn't do in Carolina."
Gourde made it 3-1 at 7:31 of the second period with a wrist shot from the high slot to finish a 3-on-2 rush.
"We didn't start good and in this league when you're chasing the game it's hard to come back," Gourde said. "We have to set ourselves up for a better start. I think we were under control a little more in the second period."
McDonagh made the score 3-2 with a one-timer from above the left circle that beat Markstrom to the stick side.
Jesper Bratt put the Devils ahead 4-2 at 4:48 of the third period, scoring short-handed on a breakaway.