Quenneville (1,000-595-153 with 77 ties) joins Scotty Bowman (1,244-573-10 with 314 ties) as the only NHL coaches to reach the milestone.
Leo Carlsson had a goal and two assists for Anaheim — including setting up Gauthier for the go-ahead goal — after missing the previous 11 games following a procedure to treat a left thigh injury.
Chris Kreider had two assists, and Lukas Dostal made 22 saves for the Ducks (31-23-3), who are 10-2-0 in their past 12 games and have won six straight at home.
Matt Savoie had a goal and two assists, Ryan Nugent-Hopkins and Evan Bouchard each had a goal and an assist, and Connor McDavid and Mattias Ekholm each had two assists for the Oilers (28-23-8), who have lost a season-high four straight.
Tristan Jarry made 20 saves before he was replaced in the third period by Connor Ingram, who had three saves.
Edmonton scored 13 seconds into the game when Savoie made a stretch pass up the wall to Jack Roslovic, who got behind Ducks defenseman Radko Gudas and scored with a snap shot from the left circle for a 1-0 lead.
The Oilers made it 2-0 at 9:03 when McDavid recovered a loose puck behind the Anaheim net and passed it out front to Nugent-Hopkins, who lifted the puck over Dostal's right shoulder from in close.
Ian Moore stepped into the puck after it banked off the side boards and scored up high with a one-timer from the right circle to cut it to 2-1 at 10:30.
Alex Killorn stopped a short-handed rush in the Anaheim zone and then put in a rebound while still on the power play to tie it 2-2 at 3:01 of the second period.
Zach Hyman redirected a pass from Ekholm into the net to move the Oilers back in front 3-2 at 18:45.
Bouchard scored 35 seconds later on a short feed from Nugent-Hopkins at 19:20 to re-establish the two-goal lead at 4-2.
Carlsson cut it to 4-3 at 1:30 of the third period when he put in a rebound off a rush. Olen Zellweger followed with a shot from above the left circle that sailed through traffic and hit the net to tie it 4-4 at 2:55.
Edmonton moved back in front 5-4 at 5:53 when Savoie slapped in a loose puck from in close while on a power play.
The Ducks came back and tied it 5-5 at 6:39 when Beckett Sennecke scored from the inside edge of the right circle off the rush, ending the night for Jarry.
