Real won 2-0 in Warsaw thanks to goals from Fede Valverde and new signing Kylian Mbappe on his debut.
Ancelotti later said: "I'm very happy. It's an honour and a privilege to coach this team and we did well today. It was a difficult game in the first half because Atalanta played very well. They were very strong in the duels and we had difficulties. But as soon as the game opened up a bit, you could see the quality we have up front.
"Mbappé has incredible quality. He worked well, combining with his team-mates and working hard in defence. The team didn't play well in the first half, but they worked very well defensively and that was the important thing. When the spaces opened up, it was easier for us. All the attacking players like Bellingham, Vini and Rodrygo contributed. The team played very well in the second half.
"Bellingham, Valverde and Tchouameni did a great job. It was a very different game, with a lot of duels and one on ones; it was difficult to look for superiority. We thought about playing more vertically, playing balls in behind. The first half didn't go well, but afterwards we had a lot of space and more chances. All three played very well, especially defensively, and gave the team balance at the back."
On Atalanta counterpart Gian Piero Gasperini, he said: "I congratulated him for the game they played. I told him that playing against Atalanta is like going to the dentist. They were very strong in the duels and, when they slowed down, we had more space and we were able to take the lead."