Nick Schmaltz cut it to 4-3 at 1:29 of the third period with a one-timer from the left circle off a pass by Clayton Keller.
McBain tied it 4-4 at 18:00 when he kicked the puck to his stick and scored past a diving Vanecek.
"Our game wasn't how we wanted to play. It was a tough, tough game at first," McBain said. "But we bounced back and it's unfortunate we didn't get the results we wanted."
Meier gave New Jersey a 1-0 lead at 9:09 of the first period, collecting a rebound in front and scoring five-hole with his backhand.
"[I had] a lot of energy to go out there and play," Meier said. "I haven't played in a bit so I tried to keep it simple. The puck just followed me in front off the rebound and, yeah, it's obviously a great way to start."
Matias Maccelli tied it 1-1 at 10:19 when he cut through the slot and scored with a one-timer off a pass from McBain.
Boqvist responded for New Jersey to make it 2-1 at 18:18, scoring with a backhand after receiving a cross-crease pass from Severson.
Severson then extended it to 3-1 on a power play at 4:44 of the second period. He scored with a wrist shot through traffic from above the right circle after receiving a back pass from Mercer.
Zack Kassian cut it to 3-2 at 16:23 when he stepped into a rebound in the right circle and scored with a slap shot. It was his second goal this season and first since Oct. 25 (38 games).
Boqvist pushed it to 4-2 at 19:19, scoring with a one-timer from the high slot off a centering pass from behind the net by Ondrej Palat.
"I've had a couple of chances to score here lately, so it was nice to put those in, for sure," Boqvist said.
NOTES:Vanecek is 10-0-1 in his past 11 starts on the road. His last loss on the road was Dec. 20 to the Carolina Hurricanes. … Devils defenseman Dougie Hamilton extended his point streak to five games (two goals, five assists). He has 18 multipoint games this season, passing Scott Stevens (1991-92, 1993-94) for the most by a defenseman in Devils/Colorado Rockies/Kansas City Scouts history. … Connor Mackey had one assist, one shot, four hits and one block in 13:33 of ice time in his Coyotes debut. He was acquired from the Calgary Flames on Friday. Brett Ritchie, who was also acquired in the trade, was a healthy scratch.