Hedjuk beat goalie Ilya Bryzaglov with a backhand shot in the fourth round of the shootout, after Wojtek Wolski had previously scored for the Avalanche and Chris Kunitz for the Ducks in the opening round.
Anaheim's Ryan Getzlaf sent the game to overtime with a goal from the slot off a feed from Corey Perry with 11:03 to play.
Rookie Curtis Glencross also scored for the Ducks in his in his first NHL game.
Andrew Brunette and Wolski scored for Colorado. Wolski had a breakaway chance to win the game on overtime, but Byrzgalov stuck out his left pad to force him to shoot wide.
Peter Budaj made 38 saves for the Avalanche, improving to 4-3-1 while starting the last eight games for Colorado.
The Ducks are now 2-4-2 since losing starting goal Jean-Sebastien Giguere (groin) and defenceman Francois Beauchemin (lacerated spleen) in a game at San Jose on Dec. 26. Five days later, they also lost defenceman Chris Pronger with a broken foot. All three players are not expected to return before the All-Star Game on Jan. 25.
Brunette skated untouched from behind the Anaheim net and backhanded the puck past Byrzgalov 4:57 into the game to put Colorado ahead.
On his second shift in an NHL game, Glencross found himself with the puck alone in front of the Avalanche net after a sharp centring pass from the right sidewall by Getzlaf. He quickly snapped a shot over Budaj's right shoulder for his first NHL goal.
Glencross, a native of Kindersley, Alberta, was just recalled from the Ducks' AHL affiliate in Portland, Maine, on Friday.
Wolski regained the lead for Colorado, swatting in a rebound fluttering in mid-air to Bryzgalov's right for his 16th goal of the season. That tied him with Ducks winger Dustin Penner for second among NHL rookies, behind Pittsburgh forward Evgeni Malkin.
Notes: The Ducks ran a satire of a Saturday Night Live skit on the video board before the game which led into their only Saturday night home game this season. The video featured injured Ducks defenceman Chris Pronger and Francois Beauchemin sitting around a restaurant table with fans talking about "Da Ducks." ... Brunette has now scored at least 15 goals in six straight seasons.