Scheifele gave the Jets a 1-0 lead with a power-play goal 3:45 into the first period. Kailer Yamamoto tied it 1-1 at 7:33 before Ethan Bear gave the Oilers a 2-1 lead with a power-play goal at 13:02.
McKenzie tied it 2-2 at 7:42 of the second period, and Scheifele made it 3-2 with a shorthanded goal at 10:01.
Blake Wheeler, who also had two assists, scored a power-play goal to make it 4-2 at 7:24 of the third period. McKenzie extended the lead to 5-2 at 10:23.
Yamamoto scored his second of the game for the Oilers to cut it to 5-3 with 40 seconds left in the third period.
Dennis Everberg and Dustin Byfuglien each had two assists, and Connor Hellebuyck made 38 saves for Winnipeg.
Mikko Koskinen, who signed a one-year contract on May 1, allowed five goals on 21 shots for Edmonton, which outshot the Jets 41-21.