Clayton Keller scored on a power play to cut it to 2-1 at 15:52 when his centering pass went in off the stick of Edmonton forward Zach Hyman.
McLeod responded with a power-play goal on a breakaway at 2:32 of the second period to make it 3-1.
Warren Foegele made it 4-1 at 12:28 on a rebound off a point shot from Tyson Barrie, and Hyman made it 5-1 on a rebound off a shot from Draisaitl at 16:11.
The Oilers outshot the Coyotes 19-12 in the second.
"It's always tough doing it back to back after travel, but that's no excuse; we've done it many times this season," Arizona defenseman Dysin Mayo said. "I think we just didn't have the intensity to come out like we have this season. We found our way back in the game and then we let in a couple more and we really couldn't recover after that."
Draisaitl scored at 9:53 of the third period, redirecting a pass from Puljujarvi for the 6-1 final. With 48 goals, Draisaitl is tied for the NHL lead with Auston Matthews of the Toronto Maple Leafs.
"It was really important that we learned a lesson from our last game," Koskinen said. "We really wanted to bounce back. It was an embarrassing game on Saturday, and I think we got exactly the result that we needed. Not just the result, but the way that we played. We didn't give any odd-man rushes, we played well. That was the most important thing, with the two points."