Giordano cut it to 4-3 at 17:07 of the second, skating in from the left boards and scoring short side on Grubauer.
The Flames outshot the Avalanche 14-6 in the third period, including a breakaway by Monahan at 2:42 that Grubauer stopped.
"We had some moments 5-on-5, and the third period is a good example of that," Calgary coach Bill Peters said. "That was our best period and it's a period we stayed out of the box. We were able to get guys moving and skating, a little rhythm to the game."
Donskoi scored into an empty net with 1:09 left in the third for a 5-3 lead.
"It felt so good to get the win right away in the first game," said Donskoi, who spent his first four NHL seasons with the San Jose Sharks. "That was fun. It felt amazing. I've been waiting for this moment all summer."