Hellebuyck earned his first shutout of the season.
The Jets snapped a two game losing streak, their first of the season.
Blake Wheeler and Paul Stastny also scored for the Jets, who moved to 19-11-2.
Thatcher Demko stopped 25 shots for Vancouver, who dropped to 18-17-3.
Blake Wheeler got the game winner unassisted 9:41 into the first period when he poked the puck past a pinching Alex Edler before skating in and beating Thatcher Demko between the legs.
There was no scoring in the second period.
Winnipeg fired three goals in the third to seal the win.
The clubs face each other again Wednesday night in Vancouver.
SHOTS-WINNIPEG 29 VANCOUVER 22
NOTES-Canucks C Bo Horvat took a shot off the foot in the third period and did not return.