Arkhipov beat Miikka Kiprusoff with a high screened shot from the right circle.
Arkhipov had deflected in Duncan Keith's shot from left wing with 44 seconds left to tie it 3-3 and set up overtime.
Chicago won for the first time since Jan. 2 in St. Louis, ending an 0-8-2 slide. The Blackhawks also won for the first time at home since Dec. 26, stopping an 0-6-2 run.
Alex Tanguay scored on a breakaway with 3:17 left in the third to give Calgary a 3-2 lead. Tanguay took a long pass from Matthew Lombardi, slipped past defenceman James Wisniewski and beat Nikolai Khabibulin with a shot high on the stick side.
Calgary's Daymond Langkow and Kristian Huselius scored power-play goals 1:49 apart late in the second period to erase Chicago's a 2-0 lead. Tanguay assisted on both goals.
Wisniewski and Jeff Hamilton also scored in regulation for the Blackhawks.
Chicago's Nikolai Khabibulin stopped 19 shots.
Kiprusoff made 36 saves in his 10th straight start.
Huselius extended his point scoring streak to nine games. He has six goals and seven assists during the span.
The Blackhawks scored the winning goal on their eighth power-play chance after failing to convert seven previous opportunities. In their last 11 games, the Blackhawks have scored just four power-play goals in 50 chances.
Wisniewski scored the only goal of the first period at 9:21, two seconds after a penalty to Calgary's Marcus Nilson expired.
Wisniewski beat Kiprusoff with a high shot on the glove side from the right circle to complete a crisp passing sequence with Tuomo Ruutu and Peter Bondra.
Hamilton made it 2-0 at 2:35 of the second. Although he was apparently covered by two Flames players at the edge of the crease, Hamilton was able to turn and slide in a forehand shot after taking a feed from Patrick Sharp.
Langkow and Huselius then tied it with their power-play goals.
Langkow connected with 8:13 left in the second, jamming in the puck from the edge of the crease after Tanguay slid it in front of the net from the right corner.
Huselius converted a rebound of Tanguay's shot with 6:24 remaining. With Khabibulin down, Huselius lofted in a high shot from the left side of the crease.
Kiprusoff made a point-blank save on Sharp at 10:20 of the third to preserve the tie.
Notes: Czech Republic President Vaclav Klaus attended the game. The United Center organist played the Czech national anthem before the contest. . . . Chicago C Peter Bondra returned after missing four games because of a shoulder injury. . . . Chicago goalie Patrick Lalime was in uniform, albeit as a backup to Khabibulin, for the first time this season, Lalime missed the first half of the season after undergoing surgery to repair a herniated disc. Chicago D James Vandermeer sat out his first game with a broken ankle. . . . Flames D Robyn Regehr missed his second game with the flu. . . . Calgary RW Chuck Kobasew also didn't skate. He played in the Flames previous contest in Minnesota on Friday, but had missed the previous eight games with a concussion. . . . Flames players still out with long-term injuries include RW Jarome Iginla (missed 10th game with a knee sprain) and LW Darren McCarty (missed 15th game with a hip injury).