Lawson Crouse, Matias Maccelli, Clayton Keller and Liam O'Brien scored for Arizona.
''(Vejmelka) made key saves at key moments,'' Arizona coach Andre Tourigny said. ''They had obviously great chances and he made it look easy at times.''
Tage Thompson scored for Buffalo, which has lost three in a row after winning three straight. Eric Comrie made 20 saves.
''Their first three goals went off of us, and don't be fooled by us,'' Sabres coach Don Granato said. ''We didn't play hard enough in the areas we need to play hard. We were too slow that we hit goal posts. Their goalie played real well. But the bottom line is we could've won a game but we didn't go attack a game like we needed to.''
Crouse opened the scoring just 23 seconds into the game following a mix-up by Buffalo defensemen Kale Clague and Owen Power. Maccelli quickly found Crouse for an easy goal from short range.
Thompson put Buffalo on the board with a power-play goal with 5:34 left in the first. Inside the left circle, Thompson cut further inside before ripping a shot high into the far corner. It was his eighth goal of the season.
Maccelli was credited with Arizona's second goal with 3:30 left in the first after his pass ricocheted off Sabres center Dylan Cozens' skate into the Buffalo net.
Arizona took a 3-1 lead 2:24 into the third off another deflection off Cozens. O'Brien's long shot from the point was deflected by the Sabres center on its way to the net.
Vejmelka's best save of the night came with 3:22 left when he stopped Buffalo's Rasmus Asplund from close range.
''You have to tip your hat to their goalie,'' Thompson said. ''He played a good game. Sometimes you run into a hot goalie or you just can't find the back of the net. Just kind of what it is.''
Keller added an empty-net goal with 2:10 remaining.
There was a 20-minute delay before the start of the third period due to a Zamboni malfunction. The Sabres kept the hometown crowd engaged by playing music and showing dancing fans on the Jumbotron. One of the loudest ovations of the entire evening came when the Sabres showed Buffalo Bills left tackle Dion Dawkins on the Jumbotron during the delay.
Sabres defenseman Rasmus Dahlin returned after missing one game with an upper-body injury. The Sabres are beginning to get healthier on the blue line; Ilya Lyubushkin returned on Saturday after missing four games.
LONG JOURNEY AHEAD
This was the second game in a 14-game road trip for the Coyotes. They opened the long trip with a 3-2 win over Washington on Saturday. Their next home game isn't until Dec. 9 against Boston.
Coyotes: At the New York Islanders on Thursday.
Sabres: Former Sabres captain Jack Eichel and the Las Vegas Golden Knights come to Buffalo on Thursday.
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