Perry wristed a shot over a sprawled Sergei Bobrovsky for his sixth of the season. A pass by Columbus' Dalton Prout behind the net caromed hard off the back boards to Perry in the slot.
The Ducks lost Saku Koivu late in the second period on a hit that resulted in the ejection of Columbus' Brandon Dubinsky.
Artem Anisimov scored twice, rebounding from a first-period hit by Francois Beauchemin, and Blake Comeau scored his first of the season for Columbus, which finished a four-game homestand at 3-1.
Bobrovsky, last season's Vezina Trophy winner, stopped 28 of 32 shots.
Ryan Getzlaf, Peter Holland and Emerson Etem scored for Anaheim.
Anaheim is playing the fourth of an eight-game road trip. The Ducks improved to 2-2 on the trip after starting the season winning seven of their first eight games. Rookie Frederik Anderson made 18 saves to improve to 3-0 on the season.
Tied at 2, the Ducks began the third period on a five-minute power play after Dubinsky was called for interference and ejected after an open-ice hit on Koivu, who had just passed the puck crossing into the Columbus zone. Koivu was helped from the ice and did not return.
Anaheim's power play took advantage with Etem stuffing home the rebound of a Teemu Selanne shot at 1:51 of the third for his second of the season.
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Columbus kept plugging away and Anisimov made it 3-3 with his fourth of the season at 8:24, redirecting James Wisniewski's point shot.
Comeau opened the scoring at 12:56 of the first period. Ryan Johansen made a nice play and threaded a pass to Comeau for a one-timer from just inside the blue line.
Anaheim pulled even 19 seconds into the second period with Getzlaf, on a nice behind-the-back pass from Perry, zipping a low shot from the right dot for his fifth of the season.
Holland made it 2-1 at 6:59, throwing an off-angle shot from near the left end line that squeezed inside the near post past Bobrovsky.
The Blue Jackets, who didn't register their first shot of the period until midway through, scored a short-handed goal for the second game in a row to get back in it.
Cam Fowler bobbled the puck and Dubinsky stole it and sent Anisimov on a breakaway. He deked Anderson to make 2-2 at 12:12.
NOTES: Columbus' next game is Friday at Pittsburgh to start a home-and-home series with the Penguins. ...The Ducks continue their road trip Tuesday in Philadelphia. ...Columbus general manager Jarmo Kekalainen and Teemu Selanne played together on Team Finland at the 1991 Canada Cup tournament. ...Anaheim is 6-0 in one-goal games.
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