Perry corralled the puck after Sami Vatanen knocked it out of the Ducks' zone and raced down the ice before sending it past Semyon Varlamov 1:25 into the extra period, securing another dramatic late-game victory.
The Ducks are now 29-1-7 in games decided by one goal this season.
Frederik Andersen stopped 38 shots for the Ducks, picking up his second win in four starts since suffering a neck injury in February.
Varlamov made 37 saves, as the Avalanche's streak of four consecutive road wins came to an end.
That success, combined with wins in six of its first seven games in March, had put the Avalanche back on the edge of playoff contention. But this loss leaves them six points back of a spot with only 11 games remaining, including six against teams currently in the playoff positions.
However, the Avalanche boosted those slim hopes with a pair of goals in the third period to claim one point.
Less than two minutes later, Ryan O'Reilly picked Vatanen's pocket on the power play and pulled away for a short-handed goal, beating Andersen on his stick side for a brief 2-1 lead.
But on that same man advantage, Rickard Rakell responded with a wicked wrist shot that sailed past Varlamov to tie the game at 2-2 with 10:56 remaining in regulation.
Jiri Sekac opened the scoring by curling around the net and putting a low-angle shot over Varlamov's left shoulder 3:37 into the first period. Sekac, Tomas Fleischmann and Rakell, who recorded assists on the play, each made aggressive moves to retain possession during the nearly minute-long barrage in the Avalanche's end.
Vatanen returned for the Ducks after missing 15 games with an undisclosed lower-body injury, playing 22:21. Vatanen leads the team's blue line production with 12 goals and 35 points.
The Ducks improved to 32-9-5 against the Western Conference, including a staggering 16-3-0 record against the Central Division.
NOTES: Centers Nathan MacKinnon (foot) and Joey Hishon (elbow) did not play. The Avalanche have lost the second-most man games to injuries in the NHL this season, trailing only the Columbus Blue Jackets. . Winger Emerson Etem and defensemen Clayton Stoner and Korbinian Holzer were healthy scratches for the Ducks. ...The Avalanche will return to Colorado before resuming their five-game road trip against the Calgary Flames on Monday. ...The Ducks will open a five-game East Coast swing against the Metropolitan Division-leading New York Rangers on Sunday.
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