Vatrano scored on the penalty kill to give Anaheim a 1-0 lead at 11:54 of the first period. Rasmus Andersson turned the puck over at the offensive blue line, and Vatrano beat Wolf with a wrist shot over the glove from inside the left face-off circle on the rush.
Klapka scored 15 seconds into the second period to tie it 1-1, skating around Brian Dumoulin before beating Gibson backhand around his right pad for his first goal of the season.
Coleman gave Calgary a 2-1 lead at 19:16 when Coronato’s shot deflected off him and in.
"I think just coming in between periods... we have a lot of guys that are vocal and great leadership," Coronato said. "I think those older guys did a great job kind of getting everyone going."
Wolf made five saves in 23 seconds in the third period, including on a one-timer by Jansen Harkins to end the sequence at 6:26 and keep the Flames ahead.
"We just couldn't keep it going," Ducks defenseman Jackson LaCombe said. "We just didn't have our right legs. We couldn't keep it going."