Minnesota led 1-0 after two periods, but Niklas Hagman tied it at 8:17 of the third and Perry swooped in to beat Josh Harding and give Anaheim the lead for good.
Dany Heatley scored his team-leading 19th goal, but the Wild couldn't generate much offense the rest of the game.
Minnesota has scored just seven goals in its last six games and is 5-16-5 since Dec. 13.
Hiller recovered nicely after allowing Heatley's early goal and is 11-2-3 in his last 16 games.
The surging Ducks are creeping up in the Western Conference standings. They won for the first time this season when trailing after two periods.
After starting the season 20-7-3, Minnesota has been slowed by injuries and offensive difficulties. The Ducks are now just two points behind the Wild for 12th in the West.
Minnesota has used an NHL-high 39 players this season, one away from the team record of 40 set last season.
Hagman's goal came after a rare Mikko Koivu turnover in the Wild's zone and was a relief for the Ducks.
Matt Beleskey had an open net early in the second period, but his shot was blocked by Marco Scandella. Ryan Getzlaf hit the crossbar at the end of the third and the Ducks failed to score while on a four-minute power play.
Minnesota failed to seize the momentum, though, thanks in large part to Hiller.
The red-hot goalie denied Heatley early in the third and withstood a flurry in front of the net as the final seconds ticked away late in the game.
Perry beat Harding with a slick wrist shot over the goalie's right glove. He has 13 goals in 2012, second most in the NHL.
The Ducks have now earned points in 16 of their last 18 games and nine of their last 10 road games.
NOTES: The Ducks recalled G Jeff Deslauriers and sent G Iiro Tarkki to Syracuse of the AHL before the game. ... Tuesday's game was the 13th time brothers Mikko and Saku Koivu played against each other. ... Only four players have played all 56 games for the Wild this season. ... Wild wing Cal Clutterbuck missed the game with a lower-body injury. ... Wild D Marek Zidlicky assisted on Heatley's goal for his first power-play assist since Nov. 8.