Lowry tied it 3-3 at 6:54, and Kyle Capobianco put the Jets ahead 4-3 at 7:16.
Draisaitl tied it 4-4 at 9:18 with another power-play goal, this time redirecting a centering pass from Connor McDavid. He has 26 goals on the man-advantage this season, the most in Oilers history.
"We are in the race to make the playoffs here. We would rather take a 1-0 win," Draisaitl said. "They played a good game, but we obviously gave them way too many chances for looks on scoring chances. They are obviously a good offensive team, so they will capitalize."
Draisaitl has 20 points (12 goals, eight assists) during a 12-game point streak. McDavid has 25 points (10 goals, 15 assists) during a 10-game point streak.
"I think if you would have asked the Winnipeg Jets if they wanted to get into a track meet with the Edmonton Oilers, I don't think that's their game plan," Woodcroft said. "I thought it was a sloppily played game in many regards both ways. I think their calling card would be a prime defending team. That's what got them the record out of the gate that they had. I think both coaches look at that game as a lot of sloppy mistakes, and in the end, they found that extra goal, they found that extra save and they found the two points and we didn't get that tonight."
Barron responded for the Jets to make it 5-4 at 10:45 of the second, and Morrissey extended it to 6-4 on a power play at 19:58.
"There were a few goals I'd really like to have back," Campbell said. "You know, a guy beat me high glove and the Morrissey one found a way to beat me from the point, and then definitely five-hole on the breakaway. You know, keep working on it. I know the results will come, but just a tough pill to swallow here."
Bjugstad cut it to 6-5 at 2:55 of the third period when Devin Shore's pass pinballed off two Jets defenders to him in front. He was acquired by Edmonton in a trade with the Arizona Coyotes on Thursday.
"I almost missed it, but it was the easiest goal I scored all year," Bjugstad said. "I was fortunate, I'll take it, but like I said, I would have loved to come up with the win. First game here and now have to move on and work towards getting better and better."
DeMelo scored into an empty net at 19:25 of the third for the 7-5 final.
Vladislav Namestnikov, who was acquired from the San Jose Sharks on Friday, had the assist on the play.
"He's a competitive guy. He's not the biggest guy out there, but he's very, very competitive," Jets coach Rick Bowness said.
"The guys were paying the price to win that game tonight, and you give those guys a lot of credit. I thought Nino [Niederreiter] had a great game playing with [Scheifele and Kyle Connor]. Those shot blocks are absolutely huge. That's how you win games. That second effort, you have to pay the price to win, especially against a great team like they have."