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The Giallorossi beat Qarabag 1-0 at the Stadio Olimpico to clinch their place in Friday's last-16 draw, with Diego Perotti grabbing the vital goal.
Chelsea's draw with Atletico Madrid meant Eusebio Di Francesco's side ended up top of Group C.
"It's a good result from a difficult game, because they had everyone in defence," Nainggolan said. "We deserved to qualify. We did not wait for others and we managed to come out on top - it's a great achievement.
He added: "Now we are all focusing on the league. We are happy but we need to continue in the league. Games like this [in Europe] can be great to play in, but we need to show that same desire even against some of the weaker sides."
On the game itself, Nainggolan admitted that Qarabag had made it difficult.
"We had to find a way around them," he noted. "They were tough but we had the right patience, believing in our methods until the end. We played in the right way. The important thing is to have qualified, now we can see what happens in the draw.
"The credit goes to everyone. Compared to last year we have lost players who did well for us, so few people were expecting us to do much in the championship - but we have been doing well both in the league and the Champions League.
"The fans are close to us again and we have earned some good results: it's a nice feeling."
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