Matt Boldy, Connor Dewar and Sam Steel scored and Mats Zuccarello had two assists for the Wild, who were playing for the third time in four nights.
Gustavsson, making his 32nd career start, had four saves in the first period, 19 in the second and 12 in the third. Many of the shots he faced were from long range. He made a nice stop on Bo Horvat's deflection on a second-period power play then got a right pad on Ilya Mikheyev's shot on the doorstep with 8:15 left in the third.
''It was really nice,'' the 24-year-old Gustavsson said of his first shutout. ''You set a few small goals along the way. First, I wanted to play an AHL game, then I wanted to win an AHL game, then a shutout, then I got a chance to play in the NHL, I wanted to win in the NHL and now I got a shutout. Now we got to look at the next step.''
Kirill Kaprizov saw his franchise-record 14-game point streak end for the Wild.
Spencer Martin had 30 saves for the Canucks, who snapped a three-game win streak while getting shut out for the first time this season.
''I don't know why we came out of the game slow,'' Vancouver coach Bruce Boudreau said. ''We should have been ready to pounce. They played last night and we didn't and we let them come to us.''
Boldy opened the scoring with a power-play goal with 3:17 left in the first, just 8 seconds after Vancouver's Oliver Ekman-Larsson was sent off for hooking. Zuccarello fired a shot from the point that Martin stopped, but Boldy put in the rebound for his 10th of the season.
''Zuccy found a lane, got it down to the net, it squirted right to me and I put it in,'' Boldy said. ''Getting that win was big for us, just momentum-wise and getting our heads back in the right space.''
The Wild outshot Vancouver 14-4 in the first period. Martin kept it close stopping Jordan Greenway on a breakaway and getting a leg on Marcus Foligno's shot on a power play.
The Canucks then outshot the Wild 19-7 in the second, but it was Minnesota who went ahead 2-0 on Dewar's short-handed goal with 3:52 left.
The Wild were killing their third consecutive penalty when Vancouver's J.T. Miller threw a pass off the boards in Minnesota's zone that Dewar intercepted. He skated the length of the ice and blew a shot over Martin's blocker for his fourth.
It was the second short-handed breakaway Martin faced in the period. Earlier, he stopped Joel Eriksson Ek on a two-man breakaway.
''They had a few chances in the first period,'' Gustavsson said. ''In the second, they came into our zone, I guess their coach probably wasn't happy with them after the first period. And then we took three penalties in a row, so they got momentum. We stood tall there in those PKs and we even scored on on the PK, 2-0 into the third it felt pretty good like we were in control of the game.''
Steel made it 3-0 when he lifted a backhand over Martin's shoulder at 8:51 of the third.
NOTES: Horvat played in his 600th NHL game. ... Martin made his fourth straight start since Thatcher Demko suffered a lower-body injury Dec. 1. ... Miller has one goal in his last 11 games but has nine assists over that span. ... Foligno played in his 700th NHL game. ... Wild C Samuel Walker played his first NHL game.
Wild: Host Edmonton on Monday night in the opener of a four-game homestand.
Canucks: At Calgary on Wednesday night before returning for a three-game homestand.
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