Seth Jarvis and Andrei Svechnikov also scored, and Brandon Bussi made 24 saves for the Hurricanes (14-5-2), who are 3-0-2 in their past five games and 7-1-2 in their past 10. Sebastian Aho and K’Andre Miller each had two assists.
Gabriel Vilardi scored two goals for the Jets (12-8-0), who had won two straight. Eric Comrie made 24 saves after it was announced earlier in the day that Connor Hellebuyck will have surgery on his knee and be out 4-6 weeks.
Staal scored 16 seconds into the first period to give the Hurricanes a 1-0 lead. He received a cross-ice pass from Jordan Martinook and skated into the right circle, where he made and inside-outside move on defenseman Dylan Samberg before beating Comrie with a wrist shot that went in off the far post.
Josh Morrissey tied it 1-1 at 3:45. Mark Scheifele skated in on a 2-on-1 rush with Kyle Connor, but he passed back to Morrissey, who was the trailer on the rush and scored with a wrist shot from the left circle.
Vilardi scored on a power play to put the Jets in front 2-1 at 9:28. He blocked a dump attempt from Carolina defenseman Mike Reilly in the neutral zone, and the puck went right to Cole Perfetti, who skated in on a 2-on-0 rush before passing back to Vilardi for a one-timer in tight.
Staal tied it 2-2 with his second of the game at 3:08 of the second period. He redirected Jalen Chatfield’s shot-pass from the right point while skating through the high slot.
Chatfield, who was activated off injured reserve earlier in the day, returned after missing seven games with an upper-body injury.
Jarvis scored short-handed to give Carolina a 3-2 lead at 6:31. He got to a rebound in front after Comrie stopped Aho’s breakaway attempt.
Svechnikov pushed it to 4-2 at 3:19 of the third, scoring a power-play goal with a one-timer from the right circle off a cross-ice pass from Nikolaj Ehlers.
Ehlers was playing his first game in Winnipeg since signing a six-year, $51 million contract with Carolina on July 3. Selected by the Jets with the No. 9 pick in the 2014 NHL Draft, Ehlers played his first 10 NHL seasons in Winnipeg.
Vilardi scored his second of the game to cut it to 4-3 at 4:13. His intended pass for Connor on a 2-on-1 rush deflected five-hole off Chatfield’s skate.
Chatfield was then assessed a minor for high-sticking with 52 seconds remaining in regulation, but the Jets couldn’t tie the game on the ensuing 6-on-4 advantage.