Ales Hemsky also scored for Edmonton and connected in the shootout to seal the victory after Sam Gagner opened the tiebreaker with a goal against Jean-Sebastien Giguere.
Garon made his third start with the Oilers, who signed him in July.
The Oilers pulled even when Cogliano parked himself in the low slot and tipped Geoff Sanderson's long slap shot past Giguere's stick for his third career goal and second short-handed. It came while teammate Steve Staios served a high-sticking penalty.
Chris Kunitz and Corey Perry scored for Anaheim (4-7-2).
The Oilers faced the Ducks for the first time since luring second-year left-wing Dustin Penner away from Anaheim with a five-year, US$21.25 million offer sheet that Ducks general manager Brian Burke declined to match. The move triggered an ugly public spat between Burke and Edmonton GM Kevin Lowe, who watched the game from adjacent suites in the press box without speaking to each other.
Penner's contract pays him $3.8 million more this season than the $450,000 he earned last year, when he was second on the team with 29 goals and set franchise rookie records for goals and points.
He did not receive a warm welcome from the sellout crowd of 17,174, most of whom booed Penner whenever he touched the puck. The booing intensified during a timeout at 9:23 of the first period when the video board showed him standing at the bench and flashed the words "Welcome Dustin Penner."
Ryan Getzlaf, who led Anaheim with five goals against Edmonton last season, sat out his second straight game because of an injured left wrist. Rob Niedermayer was sidelined for the third game due to a groin injury, and fellow forward Todd Bertuzzi missed his sixth game with a concussion.
The Oilers had limited Anaheim to nine shots until Kunitz tied it 1-1 at 11:51 of the second period with a short wrist shot past Garon's glove.
Corey Perry scored on a one-timer from the slot after Andy McDonald set him up with a pass from the right corner.
Hemsky opened the scoring at 12:41 of the first period, using Anaheim defenceman Joe DiPenta as a screen and beating Giguere high to the glove side with a one-timer.
Notes: Penner, presented with his Stanley Cup ring by Ducks owner Henry Samueli, has two goals and three assists in 12 games with the Oilers. ... Perry and Getzlaf, Penner's linemates last season, have combined for nine goals and 11 assists in Anaheim's first 13 games. Ducks coach Randy Carlyle has tried at least six left-wings on the line. ... The Oilers were 0-for-4 on the power play, and are an NHL-worst 3-for-48 with the man advantage. The Ducks have one goal in 31 short-handed situations at home. ... The Oilers have allowed at least one third-period goal in eight straight games.