Antoine Roussel scored the game's only goal to help Arizona get just its fifth win of the season (5-15-2).
''Thank God we had (Vejmelka),'' Coyotes coach Andre Tourigny said of his rookie goalie. ''Veji was rock-solid and the guys on the (penalty kill) came up big . It's fun to win the game that way because our PK had been challenged early in the season.''
Vejmelka, who hails from the Czech Republic, kept the puck after getting his second career win in 14 NHL starts.
''It was a special moment for me, for a shutout,'' said the 6-foot-4, 203-pound Vejmelka, who signed with Arizona as a free agent in May. ''An unbelievable moment for me. I'm just trying to enjoy it as much as I can right now. Unbelievable night and big points for us.
''I took a picture with (the puck) and I will give it to my father because he supported me every single game when I was a child,'' he added.
Connor Hellebuyck had 14 saves for Winnipeg, which is 1-5-1 in its last seven games. The Jets went 0 for 6 on the power play, including 0 for 5 in the third period.
''There are going to be games this year where we play much worse than that where we win,'' Jets captain Blake Wheeler said. ''I don't know if I've been a part of a game where you outshoot a team by 30 in the NHL.
''Credit to them, they got in shot lanes, they sold out, their goalie played well.''
Winnipeg forward Paul Stastny agreed with Wheeler's assessment.
''You'll have games where you don't win those games and then, there will be games later down the road that we don't deserve to win, and we will find a way to win,'' he said. ''We just have to try and stay positive with it and just keep working because no one is going to feel bad for us.''
Roussel, who was left alone in front of the net, converted a pass from Loui Eriksson past a helpless Hellebuyck with 2:01 left in the second period. It was Roussel's third of the season.
The Coyotes killed off the only penalty of the second period, a hooking penalty to one-time Jets captain Andrew Ladd to keep Winnipeg at bay. The Jets outshot Arizona 15-5 in the period.
Winnipeg couldn't beat Vejmelka, despite swarming him throughout the game. He came in sporting a 1-9-1 record and .905 save percentage.
The Jets outshot the Coyotes 17-7 in the first. Vejmelka had to be sharp on several chances.
Hellebuyck also stood tall through the first 20 minutes. The Jets killed off the only penalty of the period, a tripping call to defenseman Dylan DeMelo.
Coyotes: At Minnesota on Tuesday night to wrap a two-game trip.
Jets: Host New Jersey on Friday night in the second of a four-game homestand.
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