Blake Coleman gave Calgary a 1-0 lead at 18:03 of the first after corralling a pass from Ruzicka at the bottom of the right circle and shooting over Kuemper's blocker and under the crossbar.
"To be honest, I just saw an opening open up and just kind of wired it there," Ruzicka said.
Sheary tied it 1-1 at 19:26 with a power-play goal, scoring short side over Vladar's glove from the right face-off dot, but Mangiapane made it 2-1 at 1:35 of the second with a shot over Kuemper's left shoulder.
"The second period, we didn't generate anything," Capitals coach Peter Laviolette said. "The third period, I thought that we came out and tried to push, and when you're trying to push like that, we got exposed off the rush, so we got caught with numbers through the neutral zone. Probably should've pulled out, backed off."
Ruzicka put the Flames ahead 3-1 at 13:39 of the third when he beat Lindgren glove side on a breakaway set up by Backlund.
Mangiapane then extended the lead to 4-1 just 55 seconds later with a shot from the slot.
"They're a good team over there, so you know it's going to be a close, physical game," Mangiapane said. "We knew that kind of going into it. I think we just stuck to our system, and in the end, in the third period, I don't think we gave them much. We really just waited for them to make mistakes, and we capitalized on our opportunities."