Ladislav Nagy and Georges Laraque each had two assists for the Coyotes, who have the fewest points in the Western Conference. Pavol Demitra, Wes Walz and Mark Parrish each had a goal and an assist for the Northwest Division-leading Wild, who lost for the fourth time in five games.
Phoenix goaltender Curtis Joseph stopped Demitra's wide-open shot in the final seconds to preserve the victory.
After Walz put the Wild up 3-2 at 7:00 of the third period, Perreault answered 43 seconds later with his second goal of the night.
Phoenix went ahead 4-3 at 15:41 when Saprykin corralled a rebound of his own shot and backhanded it off goaltender Manny Fernandez's shoulder and into the net.
The Coyotes have had a disturbing tendency of falling behind early in games, and they did it again when Minnesota jumped ahead 1-0 at 1:01 of the first period.
Demitra scooped up a turnover in the Phoenix zone and fired a wrist shot past Joseph for his fourth goal of the season.
Demitra burned the Coyotes again at 1:12 of the second period, feeding Parrish from behind the net.
Minnesota has allowed the third-fewest goals in the NHL, but the Wild couldn't make the 2-0 lead stand up.
The game seemed to turn when Laraque fought with Minnesota's Derek Boogaard less than two minutes after Parrish scored.
Forty-seven seconds later, Perreault beat Fernandez to his glove side to cut Minnesota's lead to 2-1. The Coyotes tied it 2-2 on Saprykin's first goal of the game.
Notes: The Wild signed assistant coaches Mike Ramsey and Mario Tremblay to multi-year contract extensions. They have been on coach Jacques Lemaire's staff since the Wild made its debut in 2000-01. . . . Minnesota was without right-wing Marian Gaborik, who has missed 10 games with a groin injury. . . . Minnesota finishes its season-long five-game road trip at Nashville Thursday night. . . . Mike Comrie, Shane Doan and Jeremy Roenick lead the Coyotes with 12 career points apiece against Minnesota. Comrie (ankle) and Doan (back) were out Tuesday night.