John McGinn and Leon Bailey struck the goals after halftime to win the round of 16 tie 3-0 on aggregate.
Former Arsenal striker Giroud said afterwards: "We started the game well. Defensively, we did not allow them to play through our lines, and they did not create many chances.
"We couldn’t manage to play enough in attack in the first half, but we did better in the second, and during a strong spell for us, we conceded the first goal, which we should have avoided."
On McGinn's goal, Giroud praised Villa goalkeeper Emi Martinez for his long ball to create the chance.
He said: "Martínez's pass was amazing. I tried to get in front of him, but to concede a goal like that at this level is frustrating.
"It's always a pleasure to come here, to the atmosphere and the stadium, I really enjoy. Our fans came out in their numbers. We wanted to do it for them, but unfortunately, we couldn’t get a better result. I’m proud of the team, and congratulations to Villa."