Mattias Janmark and Nick Bjugstad scored for Edmonton.
Jack Campbell turned aside 32 shots for the Oilers, who dropped to 34-22-8.
Draisaitl opened the scoring 1:43 into the contest on the power play when he tapped a rebound off a Zach Hyman shot past Hellebuyck to make it 1-0 Oilers.
Mark Scheifele got the equalizer 11:34 into the first when he got his stick on a Dylan DeMelo point shot and redirected it past Jack Campbell.
Josh Morrissey made it 2-1 at 15:09 when he floated a wrister from the blue line past Campbell beating him just inside the left post.
Mattias Janmark tied it up shorthanded just under three minutes into the second period when he finished off a two-on-one with Ryan Nugent-Hopkins after the Oiler centreman snagged a pass that Kyle Connor floated in the Edmonton zone.
The goal was his 6th of the season.
Leon Draistaitl netted his second of the night at 6:12 after intercepting an errant Mark Scheifele breakout pass before moving in and lifting a backhand past Hellebuyck to give the Oilers a 3-2 lead.
Adam Lowry replied 42 seconds later after redirecting a Morgan Barron feed while driving the net.
Kyle Capobianco put Winnipeg ahead at 7:16 after he beat Campbell high glove side, converting a Blake Wheeler feed to make it 4-3.
The goal was his 2nd of the season.
Draisaitl completed the hat-trick at 9:18 on the power play when he got his stick on a Connor McDavid feed from the corner as Edmonton tied it at 4.
The goal was his 41st of the season.
Morgan Barron gave Winnipeg a 5-4 lead at 10:45 converting a Josh Morrissey neutral zone feed past Campbell after driving the net hard and sliding a backhand through his legs.
Josh Morrissey gave Winnipeg a lead they would not relinquish with two seconds remaining in the second period when he converted a Kyle Connor feed past Campbell, hammering a slap shot just inside the left post to make it 6-4.
The goal was his 15th of the season.
Scott Bjugstad pulled the Oilers to within a goal just under three minutes into the third, tapping a Devin Shore feed into a wide open net at 2:55.
Dylan DeMelo banked a puck off the side boards from his own end into an empty with 35 seconds remaining in regulation to seal the win for the Jets.
SHOTS-EDMONTON 35 WINNIPEG 39