Backlund took the puck to the net and sent a pass to Sean Monahan for a shot that was stopped by goalie Darcy Kuemper. The puck bounced into the air, and Backlund batted it down and past Kuemper to give Calgary a 5-0 homestand.
The last time the Flames swept a homestand that long was in 1997, Jarome Iginla's rookie season.
TJ Galiardi, with his first goal in 38 games, and Dennis Wideman also scored for Calgary (21-27-7), which hadn't won five in a row since March 2012.
Mark Giordano had two assists to extend his career-high point streak to nine games. He has four goals and six assists in the spurt.
The Wild cut their deficit to 3-2 on Heatley's power-play goal at 10:31 of the third. Ballard then silenced the sell-out crowd of 19,289 with the tying goal at 15:55.
On a Minnesota rush, Ballard got caught up with Wideman as he drove the net. With Wideman tangled up with goalie Reto Berra, Ballard backed away just in time to one-time a centering pass from Kyle Brodziak into the top corner.
It was back on Oct. 4, in the Flames' second game of the season in Columbus, when Galiardi - acquired in an offseason trade with San Jose - scored his first goal with the Flames. It took nearly four months for him to score again.
He finally ended the drought at 12:36 of the first period to give the Flames a 1-0 lead.
After getting a pass from Kevin Westgarth, Galiardi spun and quickly sent a backhand into the top corner. He immediately looked skyward in relief and shock.
That lead held up until 14:03 of the second period when Cooke tied it for the Wild while short-handed. Cooke burst down the left wing, and from just inside the top of the circle blasted a slap shot over Berra's shoulder.
Berra didn't start the game for the Flames. He entered with 2:28 left in the first period when Karri Ramo left with a lower-body injury.
The Flames regained the lead with 1:36 left in the second on a nifty deflection from Wideman.
Calgary surged in front 3-1 one minute into the third period when Backlund finished off a 3-on-1 rush for the Flames. The short-handed goal was the third of the season for Backlund, moving him into a tie for second in the NHL behind Boston's Brad Marchand.
While most of the season has seen Calgary equally divide up the starts between Ramo and Berra, Ramo (11-10-4) had been the better of the two in recent weeks. He had been getting the majority of playing time, and Saturday was his sixth start in seven games.
Berra had 14 saves to improve to 7-14-2. He has yet to win in regulation, with four shootout victories and three in overtime.
Kuemper made his 10th straight start for the Wild. He had 28 saves and fell to 6-4-1.
NOTES: The Flames' final three games before the Olympic break will be on the road, beginning in Montreal on Tuesday. ... Minnesota will play twice at home before the break. ... The Flames have scored eight short-handed goals, second in the NHL behind Carolina (9). ... Flames D Kris Russell (upper body) didn't play. Taking his spot was Chris Breen, who played for the first time since being recalled from Abbotsford (AHL) on Jan. 26.