Byfuglien scored 51 seconds into the first period on a breakaway after taking the puck away from Sharks defenseman Brent Burns with a poke check.
San Jose tied it 1-1 at 1:36 when Kane scored against his former team for the first time, from the left face-off circle on a 2-on-1 rush.
"I played decently but could have finished better," Kane said. "Two good teams going at it. We had won five in a row [and] we were looking to extend the streak. We've definitely gotten better as a group."
The Sharks took a 2-1 lead at 11:27 on Pavelski's power-play goal, a tip of Kevin Labanc's shot.
Ehlers tied it 2-2 at 10:32 from close range into an open net. Ehlers scored after Jones went behind the net and the puck bounced off the back boards in front to Ehlers.
"It feels good," Ehlers said of his hot streak. "I think me, [Scheifele] and [Wheeler] have created some pretty good chemistry and we're working as hard as we can to keep getting better as a line and helping the team win."
Appleton gave Winnipeg a 3-2 lead at 13:44 when he scored from close range for his first NHL goal. It was his 11th NHL game.
Donskoi tied it 3-3 at 19:54 of the second period, redirecting Kane's pass for his 100th NHL point. He ended a 13-game goalless drought.
"I think we're confident in ourselves," Wheeler said. "I think we never feel out of the game, we never feel we need to press to score goals. Do the right things over and over again and we're going to get chances. It's just a matter of making it count on your best chances."
The Sharks outshot the Jets 14-3 in the third period but were outscored 2-0.
"The third, we got outshot badly, but I think in terms of quality we would have had the best chances to score," Maurice said.