Clayton Keller added two goals and an assist, Alex Kerfoot also scored, and 10 different players recorded at least one point for the Coyotes, who improved to 8-9-2 on the road this season, which surpasses their road win total from all of last year.
Arizona has won 21 times through 41 games for the first time since the 2019-20 season.
"Everybody was playing well," head coach André Tourigny said. "Everybody played good for us, so I was happy about our performance."
Marc-André Fleury made 14 saves in relief of Filip Gustavsson, who took the loss after allowing five goals on 18 shots in 26:40 of play.
Kerfoot opened the scoring with a power-play goal 5:07 into the first period, one-timing a pass from the bottom of the right circle past Gustavsson.
Bjugstad then scored the next two goals, at 11:47 and 17:21, to give Arizona a 3-0 lead heading into the first intermission.
"It was a really good start," head coach André Tourigny said. "The entire first period, I think we played the right way, we had urgency defensively, we were stingy defensively, we played a solid period."
Keller wasted little time in the second period, giving the Coyotes a 4-0 lead 1:51 into the frame. The 25-year-old finished with three points, and now has seven points over his last four games.
Bjugstad then completed the hat trick just under five minutes later, marking his first three-goal game since March 6, 2018 when he was with the Florida Panthers.
