Ritchie scored 30 seconds into the first period for a 1-0 lead on Arizona's first shot. Jack McBain, who signed a two-year, entry-level contract March 23, had an assist for his first NHL point in his third NHL game.
Dube tied it 1-1 38 seconds into the second period when he shot short side by Sateri after picking up a turnover in the Coyotes zone.
Andrew Mangiapane put Calgary up 2-1 at 1:16, Tkachuk extended the lead to 3-1 at 2:18 when he shot a no-look, backhand pass from Gaudreau behind the net, and Dube made it 4-1 at 3:11.
"We're a fragile group right now," Tourigny said. "Our confidence is low. We're happy about our first period and we get into the second and they scored two quick. Then it's three. Then it's four. That's really tough mentally. That's something we need to address. We need to be better. It's not a lack of will, of desire of our players. Right now we beat ourselves up mentally and we have a tough time getting out of those slumps and those quick goals. Instead of bouncing back we beat ourselves up."
The goals, coming on four consecutive shots, were the second-fastest four-goal run in Flames history at 2:33, behind a 1:21 span on Feb. 10, 1993, against the San Jose Sharks.
"It was exciting," Gaudreau said. "Not only for us, I think the fans were fired up. We were fired up. It's still only a three-goal game. We needed to keep pushing [and we] kept pushing. A lot of guys contributed tonight. We did a great job."
Blake Coleman gave the Flames a 5-1 lead at 15:36 when he shot five-hole on Vejmelka after taking a cross-ice pass from Mikael Backlund, and Lindholm made it 6-1 at 18:58 with his 39th goal of the season.
"Second period, we just took over the game," Tkachuk said. "It seemed like everything was going in for us. The fans enjoyed it a lot, and so did we. They were really into it tonight. It gave us a lot of momentum. We were losing, and that's huge for us."