Cam Talbot made 33 saves and Oscar Klefbom had a team-high five shots on goal in the loss.
"You've got to at least attempt to out-work the other team and we had none of that today," said Oilers Head Coach Todd McLellan.
Edmonton went 0-for-2 on the power play and 1-for-3 shorthanded.
"Other than video and personnel and systems and practice time, there's nothing else (to do to improve the special teams)," said McLellan. "We're trying to adjust or tap into those areas but those are the four areas that you have. That's it."
Winnipeg jumped out to a two-goal lead in the opening 20, scoring just 2:34 into the game.
The Jets' fourth line developed a 3-on-1 in the Oilers zone, with former Oiler Matt Hendricks feeding Marko Dano at the side of the net. Dano deposited the puck into an open cage, giving the Jets an early 1-0 lead.
"The first one tonight just completely brings us down and we're not able to fight back or create momentum against it," said Lucic.
The Oilers almost squared things up when Jesse Puljujarvi pulled a spin-o-rama backhand pass to Lucic on a 2-on-1. The winger couldn't get a handle on the puck, keeping the Jets in the lead.
The visitors kept it coming as captain Blake Wheeler got in on the scoring at 13:16. Matt Benning was attempting skating the puck out of danger in his own zone when his pocket got picked by Kyle Connor. Connor made a pass to Wheeler all alone in front of the Oilers net. He put it past Talbot forehand to put Winnipeg up by two.
"Our attention to detail and all that stuff hasn't been there like last year," said Lucic. "It's just breakdown after breakdown."
McLellan shuffled his lines to open the second period - including reuniting Patrick Maroon, Connor McDavid and Leon Draisaitl - but it was the Jets who continued the scoring with two more markers.
Kyle Connor made it 3-0 while Ryan Nugent-Hopkins was in the box for holding. Patrik Laine released a shot that Connor was able to deflect in front of Talbot. The goal was Connor's 13th of the season and 24th point.
"I think we just had a chip on our shoulder last year where no matter what happened, nothing was going to faze us or bring us down," said Lucic, reflecting on the difference between 2016-17 and 2017-18. "It's been a bit different in that sense where we've been a lot more fragile."
Bryan Little then added salt to the wound with only 2.5 seconds remaining in the period. Josh Morrissey dished the puck to Little at the top of the crease, giving him an easy goal to make it 4-0 Jets.
In the third, Zack Kassian was given a double minor for roughing and the Jets took advantage of the opportunity. Mathieu Perreault slammed home a rebound at 7:26 to make it 5-0.
"It's up to us," said Lucic about turning the season around. "We've got to find a way to hold each other accountable and not be sensitive about it. Be a little bit more thick-skinned and go out there and start having fun again."
The Oilers first game of 2018 is against the Los Angeles Kings at Rogers Place on Tuesday. Puck drop is at 7:38 p.m. MST.