Now, the Avalanche are on to the second round of the playoffs for the first time in 11 years.
Gabriel Landeskog also scored, and Nathan MacKinnon and Tyson Barrie each had three assists as the Avalanche won a playoff series for the first time since 2008. Philipp Grubauer stopped 28 shots and a penalty shot by Johnny Gaudreau in the first period.
''It's awesome to win a round,'' said MacKinnon, who had three goals and five assists in the series. ''It's been a while for this organization and I, personally, have never won a round so I'm really excited about it.''
T.J. Brodie scored for the Flames, and Mike Smith finished with 27 saves.
Calgary, which finished first in the Western Conference at 50-25-7, was beaten by a team that sprinted the final two weeks of the regular season to capture the West's last wild card and then won four straight in this series after losing the opener.
''Obviously we were the big underdogs and no one picked us to win,'' MacKinnon said. ''I didn't expect to win it in five, but we believed in this group.''
The Flames' ouster marked the first time both conferences top seeds were eliminated in the first round. East-leading Tampa Bay, which won the Presidents' Trophy with the league's best record, was swept by Columbus.
Calgary ranked second in goal scoring in the regular season behind the Lightning and tied with San Jose. However, after opening with a 4-0 victory in Game 1, the Flames didn't score more than two goals in any game. The NHL's highest scoring team on home ice this season mustered only one in Game 5.
''Some of the strengths we had during the regular season weren't strengths in Round 1,'' Flames coach Bill Peters said. ''I can't explain it to you why some of those went away. It'll be interesting as we reflect and look back and try and find some answers. ... We tried to change some things lineup-wise, we tried to change some matchups a little bit on the MacKinnon line and we didn't have an answer.''
Colorado was the more confident team after coming from behind to win both the second and fourth games of th e series 3-2 in overtime, and decisively beating Calgary 6-2 in Game 3.
In four straight wins to take the series, the Avalanche beat the Flames to the puck in all three zones and forced Calgary to chase and defend. MacKinnon was the catalyst in the Avalanche's superior speed with and without the puck.
''For anyone who doesn't think MacKinnon is not one of the best if not the best, they might want to look at this series,'' Flames captain Mark Giordano said. ''He really took control and led their team.''
Gaudreau, Calgary's top goal scorer, was limited to one assist in the series.
His penalty shot attempt in the first was unsuccessful, he shot the puck wide on a clear breakaway in the period, and had what would have been his first goal of these playoffs waived off for goaltender interference in the second.
''I had a ton of chances tonight,'' Gaudreau said. ''I didn't find the net when I should have a couple times during the series.''
Rantanen's wrist shot over Smith's left shoulder 57 seconds into the third period deadened Scotiabank Saddledome as Flames fans began coming to grips with team's impending first-round upset.
''The season we had we were not expecting to go out in five games obviously in the first round,'' Giordano said. ''Right now it doesn't feel good. We had a great season and we couldn't find a way to get leads in the playoffs and stick to it when we did have the lead.''
Colorado scored first when Landeskog deflected a shot by Barrie by Smith's glove at 9:40 of the opening period.
When Gaudreau shot the puck wide on a breakaway, the Avs scored on the return trip to make it 2-0 at with 4:22 left in the first. Rantanen circled out from behind Smith and put the puck off the back of the Calgary goalie to score.
Brodie scored with 5 seconds left in the first to pull Calgary to 2-1.
Wilson earned his first at 6:52 of the second when Rantanen blocked a clearing attempt by Calgary's Rasmus Andersson and dished to him in the high slot.
From the slot, Wilson redirected MacKinnon's shot on Smith stick-side for a power-play goal with 5:13 left in the middle period.
NOTES: Playoff veteran James Neal was a healthy scratch for the Flames with Austin Czarnik drawing in on the third line for his NHL playoff debut. ... Avalanche C Derick Brassard (illness) and D Sam Girard (upper body) were Colorado's scratches.
Avalanche will face the winner of the Vegas-San Jose series, which the Golden Knights lead 3-2.
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