Michael Grabner, Phil Kessel and Jordan Oesterle scored, and Darcy Kuemper made 49 saves for the Coyotes, the No. 11 seed in the Western Conference. Oliver Ekman-Larsson had two assists.
"It's huge for the organization," Arizona coach Rick Tocchet said. "You're always trying to chip away at certain things. Three years ago, we said we wanted to chip away at playing some important games, and this was probably the biggest, important game this franchise has had since.
"The resiliency of a lot of things that's been thrown at the organization and the team ... I'm just proud of the guys the way they dealt with it."
The Coyotes will play the Colorado Avalanche, Vegas Golden Knights, St. Louis Blues or Dallas Stars in the Western Conference First Round. The Avalanche (2-0-0), Golden Knights (2-0-0), Blues (0-2-0) and Stars (0-2-0) are playing in the round-robin portion of the Qualifiers to determine the top four seeds.
"We're all feeling pretty good," Kuemper said. "Anytime you can win a series, it's a big accomplishment. We played a really challenging team; they got so much skill over there. But we rose to the occasion and were able to win the series. We're going to enjoy it and start focusing on the next round."