Peter Forsberg had three assists and Theodore had 34 saves for the Avalanche, which clinched the No. 6 seed with the win. Josh Harding had 33 saves for the Wild, which came into the game having clinched the Northwest Division title and the No. 3 seed.
The teams have met once in the postseason, in the first round in 2003, when the Wild rallied from a 3-1 series deficit to beat Colorado in seven games.
The Avalanche needed at least a point to move into the No. 6 spot. It looked dicey when Minnesota took a 2-1 lead into the second intermission, but Sakic and Milan Hejduk scored power-play goals 3:14 apart in the third to give the Avalanche a 3-2 lead.
Brian Rolston tied it with 3:21 left in regulation to send the game into overtime.
Keith Carney scored his first goal in more than a year, and Mark Parrish also scored for the Wild.
The Wild took a 2-1 lead late in the second period when Carney, who had 10 assists in 60 games this season, took a shot from just inside the blueline that beat Theodore. It was Carney's first goal since April 5, 2007, a span of 62 games.
The Avalanche got the equalizer 18 seconds after Pierre-Marc Bouchard went off for holding. Forsberg sent a diagonal pass across the front of the net to Sakic, who gathered the puck and then put a shot over Harding's right shoulder with 14:12 left in regulation.
Parrish gave the Wild the early lead, scoring a power-play goal with 2:58 left in the first period. Parrish whacked at a loose puck near Theodore's left pad and knocked it in to make it 1-0.
Stastny tied it 5:25 into the second period. The Avalanche moved the puck around the Minnesota end, and Forsberg sent a pass to the front of the net from the left boards. Stastny lifted a shot over the pads of Harding to tie the game.
Notes: Harding's last start was March 24 against Edmonton. He has lost six straight starts and hasn't won since Feb. 10. ... The Avalanche finished with 27 wins at home, its most since winning 28 in 2001, when the club won the Stanley Cup.