Sakic scored his team-leading 35th goal and added three assists to lead Colorado past Calgary 4-3, snapping the Flames' six-game winning streak.
Colorado hasn't missed the playoffs since the 93-94 season when the franchise was based in Quebec. The Avalanche have gone 13-1-2 over the past five weeks to get within five points of Calgary for the eighth and final Western Conference playoff spot.
"It's come together at the right time and the guys are playing well, playing hard, we're enjoying this streak right now," Sakic said. "Then again, we know it's still an uphill battle.
"We've got to win all our games and hope that they lose."
Each team has three games remaining, including a head-to-head matchup in Denver on Sunday, the final day of the regular season.
"Joe was great," Avalanche coach Joel Quenneville said. "He's been like that all year, his consistency has been tremendous, his leadership is as great as there is in the game.
"I can't say enough good things about him."
"We realize that when we've had success, we've had really good starts, we've been able to draw penalties, and we've stayed out of the box," Calgary captain Jarome Iginla said. "Tonight, we didn't really do any of those things."
Stastny broke a 2-2 tie and put the Avalanche ahead for good at 14:38 of the second as Colorado took advantage of a lengthy two-man advantage.
Sakic's shot hit Andrew Brunette and went to Stastny, who was alone at the side of the net and scored his 27th goal of the season.
Sakic's goal at 14:41 of the third put the Avalanche up 4-2 and proved to be the game-winner. Langkow got Calgary within 4-3 with a power-play goal at 17:21.
Flames forward Alex Tanguay gave credit to Sakic, his teammate for six seasons up until this season.
"You can't give Joe Sakic time and space like we did," Tanguay said. "We gave him 2-on-1s, a 5-on-3 power play, you give a guy with that much talent those kinds of chances, it's going to be in the back of your net and that's what happened."
The Avalanche tied it at 2-2 at 12:02 when Sakic carried the puck into the Flames zone and centered a pass that Richardson deflected into the top corner.
The Flames led 1-0 at 18:33 of the first, turning a mistake by Budaj into Calgary's 15th short-handed goal of the season.
Budaj couldn't control a 130-foot shot from Robyn Regehr and the puck bounced into the slot where Friesen raced in and sent a slap shot into the top corner.
Sakic credited Budaj for overcoming that mistake, noting that longtime teammate Patrick Roy had the ability to forget about bad goals.
"Peter's a great goalie and the best goalies in this league don't worry about a bad goal or a bad game," Sakic said. "I played with one for nine years.
"You're going to have your bad moments and as long as you forget about it and get ready for the next one, you'll be all right."
Still on the power play, Colorado tied it on Clark's goal with 12 seconds left in the period.
Budaj finished with 18 saves and is unbeaten in regulation in his past 14 games (12-0-2).
- Sakic's goal was his 609th of his career, moving past Dino Ciccarelli into 15th spot on the goal-scoring list.
- Returning to the Calgary lineup was rookie LW David Moss and D Dion Phaneuf.
- Iginla picked up his 338th and 339th assists, moving him past Gary Suter and into third place on the Flames' list. Iginla trails Theo Fleury (460) and Al MacInnis (609).