The Devils responded with four unanswered goals in a 6:30 span in the second period.
Toffoli made it 3-1 at 4:31, scoring from the bottom of the right circle after Meier's initial shot caromed to him.
Meier then pulled New Jersey to within 3-2 at 5:19, tapping in a backdoor pass from Hughes for his first goal of the season.
"It was a roller-coaster game, but you have to earn momentum back, and we didn't do a good enough job," Meier said. "We just let that momentum swing get away from us."
Nico Hischier tied it 3-3 at 6:42, absorbing a check from Tom Wilson in order to score from near the left post.
Toffoli put New Jersey in front 4-3 at 10:01 of the second, but Strome answered back with a power-play goal 18 seconds into the third period to tie it 4-4.
"Our power play stepped up right out of the gate to sort of set the tone for the third period, and then I thought for the first half of the third we were on it," Carbery said. "Then we did enough good things in the final 10 minutes to hang on. We made smart decisions, smart reads, and smart puck plays to keep everything to the perimeter."