The Hurricanes have dropped four straight games since Paul Maurice was fired and Kirk Muller took over as coach, and have lost seven straight overall. They are in Edmonton on Wednesday night as their four-game trip continues.
The Flames are 4-1-1 in their last six games.
Morrison returned to the lineup to replace Curtis Glencross, who was out with the flu. He had been a healthy scratch the past five games.
The 36-year-old Morrison skated on a line with Iginla and Olli Jokinen, and took advantage of the opportunity.
The trio was Calgary's best line all night, starting with their first shift when Morrison set up Jokinen, who was robbed by Cam Ward. Morrison and Jokinen later combined to set up Iginla in front for his eighth goal.
Calgary took a 3-0 lead 42 seconds into the second period when Morrison capped off a 3-on-2 rush, batting in a loose puck from the edge of the crease.
Morrison capped off his first two-goal game since Oct. 26, 2010, with a pinball-style goal 1:22 into the third period, which put the Flames ahead 5-2. His slap shot from off the wing deflected off Carolina defenseman Jamie McBain, then went off the far post and Ward's leg before trickling into the net.
Morrison had offseason knee surgery and has been hampered by the injury on and off this season. He entered with one goal and no assists in 10 games.
Tlusty got Carolina on the board at 5:39 of the second. Tlusty was awarded the penalty shot after diving defenseman T.J. Brodie wiped him out as he broke in alone on Miikka Kiprusoff. Given a second chance, he faked a shot, swung wide and fired a shot under the crossbar to make it 3-1.
The Hurricanes got within one at 13:16 when Brent hammered in Staal's rebound for his first goal in 18 games.
Calgary regained the two-goal cushion 50 seconds later. Bourque stripped the puck from Derek Joslin at the Flames' blue line and charged the ice, rifling a shot through Ward's legs for his team-leading ninth goal.
Ward was pulled after Backlund's power-play goal at 10:57 in the third made it 6-3. Ward stopped 26 shots.
Backlund's score was the Flames' third power-play goal at home this season and ended a 0-for-29 run with the extra man.
Carolina mounted a comeback in the third period, scoring three times in the last seven minutes of the game. But there was only 5 seconds left after Staal's second of the night got the Hurricanes within one.
NOTES: Carolina D Joni Pitkanen (upper body) left the game in the second period and did not return. ... With Calgary G Henrik Karlsson on injured reserve, Leland Irving, who was called up from Abbotsford of the AHL on Monday, dressed as the backup. ... Carolina also had a new backup in uniform with Mike Murphy recalled from Charlotte of the AHL to replace Brian Boucher (lower body), who was injured Monday in practice. ... Kiprusoff had stopped the last five penalty shots dating to Vernon Fiddler's goal for Nashville on Nov. 6, 2008. ... Calgary's previous power-play goal at home was scored by Bourque on Oct. 26 against Colorado.