Mikko Koskinen made 19 saves for the Oilers, who are 2-3-0 in their past five games after going 7-2-1 in their previous 10.
"I thought we stayed patient till we got our energy back and really started to play," Oilers coach Ken Hitchcock said. "We played a patient, smart hockey game and then the big boys took over in the third period, which is kind of how you draw it up."
David Savard scored for the Blue Jackets, who have lost three straight (0-2-1). Joonas Korpisalo made 18 saves.
Savard scored at 11:35 of the second period to give the Blue Jackets a 1-0 lead. He got behind the Oilers defense and tipped in a centering pass from Markus Nutivaara.
Brodziak tied it 1-1 at 13:17, taking a pass from Joseph Gambardella and scoring on a shot from the left face-off circle. It was Gambardella's first point in the NHL in eight games.
"It was kind of like a chess match where neither team was taking any chances," Brodziak said. "But once we got the 2-1 goal in the third they were the team that had to take the chances and we were able to capitalize after that."
Kassian scored 45 seconds into the third period to make it 2-1, converting a pass from McDavid on a 2-on-1 rush against defenseman Seth Jones.
McDavid scored at 14:02 to make it 3-1. He took a pass from Draisaitl and his shot hit the crossbar and bounced in off Korpisalo. The assist was Draisaitl's 300th NHL point.
"Them getting one early in the third was tough," Blue Jackets forward Matt Duchene said. "It's hard for me to put my finger on what is going on right now. We had some good looks but finishing and creating that offense is a problem for us right now. We have to figure out what's going wrong. I wish I had answers."
Nugent-Hopkins scored on the power play at 16:15 for the 4-1 final. It was his NHL career-high 25th goal of the season.
"It was a pretty defensive effort from the start of the game on, there was [seven] shots total in the first period and you don't see that too often," Nugent-Hopkins said. "It was definitely tight-checking and neither team gave up much defensively. We just stuck through it and stuck with it and had a good third period and came out with the win."