Kevin De Bruyne’s 76th minute strike against Paris Saint-Germain was enough to defeat the French champions and give City a 3-2 aggregate victory to take them into the final four of Europe’s elite competition for the first time.
Pellegrini reflected: "It’s a very great achievement for the club - not only to reach the semi-final and be one of the best four teams in Europe but also to have done so with our style of play.
"A lot of people thought that we would try for a draw here but from the beginning, we thought and played as a big team and I am very happy to continue with our style.
"It’s very important to be in the semi-final because it was my job to go further in the competition. This year, the team has improved by winning the group then playing very good performances against Dynamo and a very strong Paris Saint-Germain.
"We can be the best team in Europe. It’s not easy because there are many strong teams in Europe but the target of this club is to win the Champions League.
"I know that we receive a lot of criticism but we are improving every year so this is the most important thing.
"If you are in the semi-final, you have to play one of the big teams of Europe but if we play the way we played against PSG, then we can have a lot of trust against any team."
The Chilean was delighted with the impact of Belgian midfielder De Bruyne, who also netted in the 2-2 draw in the French capital.
"Kevin has an important role for the team," he added. "He’s a very dangerous player and he did it all very well.
"I always say that good players playing together will be more dangerous so when Sergio Aguero, David Silva, Kevin De Bruyne and Yaya Toure are all on the pitch, of course the team improve."
Pellegrini was also impressed by centre backs Nicolas Otamendi and Eliaquim Mangala, who reduced the likes of Zlatan Ibrahimovic and Edinson Cavani to long-range efforts at the Etihad.
"I’m very happy for them because in some performances they have been criticised but if you review their performance across the whole season then you will see they have played well," he said.
The draw for the semi-finals of the Champions League will take place on Friday at 10:30am BST.