Two goals in the opening 11 minutes gave Manuel Pellegrini’s side the perfect platform to see out a vital 3-1 victory in Sevilla, with strikes from Raheem Sterling, Fernando andWilfried Bony put them to the top of Group D, qualifying for the next stage with two games to spare.
While progressing from the group stage has been a major hurdle for City in the past, last season they made it through only to face Barcelona in the first knock-out stage and duly made a swift exit.
With that in mind, Kompany is keen for them to build on their progress so far and top the group in the hope of an easier draw.
"The first feeling is 'Finally'," he said on BT Sport 2 as City could enjoy a truly special performance in Europe.
"It's a difficult group and for us to qualify with two games to go is an incredible achievement.
"We need to go on. It would be a first for us to finish first in our group and that's important because we have been unlucky in the draws before."
They are now one point ahead of group rivals Juventus, who drew at Borussia Monchengladbach in the other Group D contest.
Manager Manuel Pellegrini changed his team's shape by leaving out De Bruyne in order to feature Fernando, Fernandinho and Toure together in midfield, and enjoyed the results after revealing he had not been impressed with some of City's recent displays.
Pellegrini said: "I'm happy with the way we played, we scored three goals, missed three of four clear chances to score and we defended well.
"I was not happy with the way we were playing in the previous games, so I'm happy with our form in this game.
"The three midfielders - Yaya Toure, Fernandinho and Fernando - played well, pressed higher and our attack was really prepared. We played with one touch and moved into space, that was the reason we created so many chances."
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