Darnell Nurse scored on a shot from the point through traffic at 13:55 to make it 3-2, and Draisaitl scored on the power play at 18:50 off a pass from McDavid for a 4-2 lead.
"We talked a lot of how we want to play as a group and try to be consistent with it," Oilers coach Dave Tippett said. "I thought tonight's game got a little scrambly at times, but I like the way we hung in there and found a way to win. We have to keep building our group, our players are really committed to getting better every day and every game is new challenge for us, and when you have a set of three games there are challenges within that, that you have to overcome, and I thought our guys did a good job of that."
Ennis scored at 5:05 of the third period to make it 5-2, capitalizing on a turnover in front by Stutzle.
Gaetan Haas scored at 15:48 for the 6-2 final.
Edmonton has outscored Ottawa 34-15 in the seven games, and McDavid has scored 16 points (three goals, 13 assists).
"We didn't underestimate them, we wanted to play a solid game, and I think those were three pretty good games against them," Haas said. "We had solid [defense] in the [defensive] zone and tried to get the puck up as fast as possible to the offensive zone, and that's how we won."
NOTES: Draisaitl is second in the NHL to McDavid with 44 points (15 goals, 29 assists). ... Dominik Kahun scored eight seconds into an 8-5 win against the Senators on Jan. 31, and Wayne Gretzky scored eight seconds into a 9-4 win against the New York Rangers on Dec. 14, 1983, for the fastest Oilers goal to start a game. ... Two Oilers have reached 50 points in fewer games than McDavid. Gretzky did it seven times (18 games, 1983-84; 19, 1984-85; 20, 1987-88; 22, 1986-87; 23, 1985-86; 23, 1982-83; 23, 1981-82) and Jari Kurri did it twice (25, 1984-85; 25, 1983-84).