Mbappe opened the scoring with another stunning goal before Giroud, who again showed coach Didier Deschamps he is more than a third-choice forward, sealed the world champions' win with a header.
France, who travel to Denmark on Sunday, climbed to third in Group 1 with five points from as many games.
"We did what we had to do," the 1998 world champion said. "Our intentions were good, everyone defended a lot, there are a lot of positives. The mindset was good, it was a very good performance."
The hosts, who had lost two and drawn two of their four previous games, came out at top speed at the Stade de France.
Mbappe fired a shot into the top corner in the second minute, but the effort was disallowed for offside.
Aurelien Tchouameni's attempt from just outside the box after a quick combination between Antoine Griezmann and Mbappe was tipped over by Austria keeper Patrick Pentz as Les Bleus piled on the pressure.
Defender Kounde became the latest casualty in an injury-ravaged squad when he limped off the pitch with a muscle problem and Maignan was replaced by Alphonse Areola at halftime due to a calf pain.
Mbappe, denied by Pentz after a one-two with Giroud, tormented the Austrian defence with his pace and created space for his teammates, Jonathan Clauss just missing out on a goal early in the second half.
Mbappe broke the deadlock with a cool finish after beating five defenders following a perfect pass from Giroud.
It spurred Austria into action, but they lacked accuracy in the box and it was France who scored again.
Giroud found the back of the net in the 65th minute from Griezmann's pinpoint cross for his 49th international goal, two shy of Thierry Henry's national record, while also becoming the oldest player to ever score for France.
The AC Milan striker was replaced by Ousmane Dembele to the Stade de France crowd's standing ovation, leaving Deschamps, who has only called up Giroud in the absence of Benzema, with a tricky decision before November's World Cup.
"Oliver deserves it, I am very happy for him. He had difficult periods with his club but always performed well with us. If I call him, it's so he can score goals. And he had many opportunities," Deschamps said after the match.
However, the French coach urged caution as it regarded his decision to include Giroud in his final squad for Qatar, where they will begin against Group D play against Australia on Nov. 22.
Deschamps said: "Do not put words in my mouth. I did not say that he would be there. There is competition for every position and I can only take 26 players."