Keller made it 3-1 at 2:50 of the second, beating Ullmark from inside the left face-off circle, and Carcone cut it to 3-2 at 3:22 after taking the puck away from Bruins forward Matthew Poitras.
"I think we just had to stay calm," Carcone said. "There was obviously some frustration going on. I thought maybe it was taking away from us, but [Tourigny] reassured us that we're playing a good game, and we just got to stick with it. I think we responded well to that."
Danton Heinen made it 4-2 at 9:12, taking a pass from Poitras and scoring from the slot.
"[Poitras] won a battle there on the wall," Heinen said. "First good reload by him, and then you know, a little chip there to me. I was lucky enough to put it in, but that was a good play by him and [Frederic] there to create all that."
Crouse cut it to 4-3 at 3:20 of the third period from the slot.
The Coyotes had two power-play opportunities after after that but failed to capitalize. They finished 0-for-5 with the man-advantage.