The Bruins outshot the Coyotes, 12-7, in the first period, but Arizona led 2-0 when it was over.
Schmaltz gave the Coyotes a 1-0 lead when he scored a power-play goal off the left post and past Tuukka Rask at 12:22. The goal was Schmaltz's fourth in seven games since coming to Arizona in a trade with Chicago on Nov. 25. All four goals have been power-play goals.
Cousins upped Arizona's lead to 2-0 just 40 seconds later when he beat Rask with a wrist shot from the top of the right face-off circle.
But the game turned as it neared the middle of the second frame. Danton Heinen started Boston's furious comeback when he scored a backdoor goal on Darcy Kuemper at 8:40. David Pastrnak tallied nearly the same goal just 33 seconds later, and Boston's blitz was on. Brad Marchand capped it by scoring a pair of goals within a span of 1:59.
"If you give them odd-man rushes all night, they've got some high-end skill and they're going to make you pay," Schmaltz said.
• Kuemper returned to the net after nearly a three-week absence because of a lower-body injury and stopped 22 of 26 shots.
"I was pretty happy with a lot of parts of my game today," Kuemper said. "A few things, just some reads and timing, I've got to watch the video and get a little bit better at, but overall I felt pretty good in there."
Kuemper made his best save when he blocked a short-handed breakaway shot by Marchand in the first period.