The Jets scored just over a minute into the contest, building a 3-0 lead before the Golden Knights were able to respond with a goal of their own.
Patrik Laine and Joel Armia scored for the Jets.
Blake Wheeler had three assists for Winnipeg.
Brayden McNabb scored for the Golden Knights.
Marc-Andre Fleury turned aside 22 shots for Vegas.
Dustin Byfuglien gave Winnipeg the lead just over a minute into the opening period.
The Jet defenceman took a drop pass from Mark Scheifele just inside the Vegas blue line which he promptly fired past Vegas goaltender Marc-Andre Fleury, beating him blocker side to give the Jets a 1-0 lead at 1:05.
With Alex Tuch off for hooking, the Jets power play gave them a 2-0 lead as Blake Wheeler found Patrik Laine unguarded and hit him with a crisp, cross ice pass which he one timed past a sprawling Marc-Andre Fleury.
Wheeler and Paul Stastny exchanged passes as Laine moved in from the blue line on Fleury's right side before the Jets captain hit him with a cross ice pass which the second-year Finn promptly fired past Fleury.
The goal was the 4th of the playoffs.
"It's obviously huge to get the first goal," said Laine. "I think it's just everybody maybe feels a little bit looser on the ice. It's hard to say. For me, it gives me a confidence a lot, because we're in the lead and it's easier to play."
Winnipeg went 3-0 when Joel Armia redirected a puck past Fleury at 7:35 of the first period when he was cutting through the crease and knocked the goaltender's stick out of his hand.
Referee Kelly Sutherland ruled that he had interfered with Fleury and disallowed the goal but Jets Head Coach Paul Maurice challenged the ruling that the goaltender had been interfered successfully and the Jets went up 3-0.
The goal was the 2nd of the post-season.
Brayden McNabb got the Golden Knights on the board 35 seconds later when he converted a Jonathan Marchessault feed by wiring a wrister past Hellebuyck at 8:10.
Marchesseault took control of the disc behind the Jets net before hitting McNabb with a crisp forehand pass at the top of the left faceoff circle which he promptly fired past Hellebucyk, beating him low glove side.
The goal was his 2nd of the playoffs.
"A disallowed goal turns into a goal," said McNabb. "It's unfortunate but it happens. We reset it, we were able to get one right after." With McNabb off for hooking Mark Scheifele tipped a puck past Fleury to give them a 4-1 lead.
Dustin Byfuglien shot the puck which Scheifele got his stick on from about 20 feet out as the disc went underneath Fleury's glove hand.
The tally was his 12th in 13 playoff games this season.
William Karlsson cut into the Jets lead at 15:55 of the 2nd, tipping a hard Jonathan Marchessault feed from the point past Hellebuyck as Vegas capitalized on a power play to make it 4-2.
The goal was his 5th of the post-season.
“Not the start we wanted and they just played two days ago, but they were game ready and we weren’t. We’re going to see what kind of team we are. (Monday) is definitely a must-win game. As a group, as long as you move on, everyone needs to step up if you want to win against the best team in the league. Their push was good, but we knew there were going to come. We didn’t get as many shots on goal as we wanted, but most of that is because of the first period. We were better in the second and third, but they played real well with the lead and that was hard for us," said Marchessault.
There was no scoring in the third period.
The Golden Knights weren't able to mount much of an offence at all in the final frame, not registering their first shot-on-goal until past the 17 minute mark.
Vegas Head Coach Gerard Gallant was honest in his assessment of his club's performance. "We chased the game all night and they (Jets) played a real good game . They played a great defensive game. They created offensively too."
Game 2 goes Monday night in Winnipeg.
SHOTS-LAS VEGAS 21 WINNIPEG 26