City were staring down the barrel of elimination after collecting just two points from their opening four matches in Group E.
The Premier League champions gave themselves hope with a late resurgence to beat Bayern Munich 3-2 in matchday five, though, and they secured their place in the last 16 with a 2-0 win at Roma on Wednesday.
Milner believes City's accomplishment of reaching the knockout stages is even more impressive given a host of injuries this season, including star striker Sergio Aguero who was absent in the Stadio Olimpico with a knee injury, and feels the team will be a match for Europe's elite.
"We will take it step by step but I definitely think the quality is there to go far in this competition," Milner said.
"We don't want to get ahead of ourselves, but the more seasons we play in the Champions League, and the more games we play in it, the experience helps and we get better as a team.
"The pressure was there because we had to win the last couple of games, and we had a couple of injuries, but that shows the quality in the squad, as we managed to get the result we needed.
"This team responds when we need to win games - we did it last year in the Premier League and now we have done it again in the Champions League, by winning these two games.
"We had a bit of help from the other game, but to beat Bayern Munich and Roma to qualify, and do it with a number of injuries, shows the amount of quality there is right through the squad."
With City finishing runners-up in the group, they face a tricky last-16 encounter and Milner somewhat tongue-in-cheek suggested that the draw is likely to throw up another meeting with Barcelona - who knocked City out at the same stage last season.
"We like drawing the same teams, and we like getting tough draws so it will probably be Barcelona again," he added.
"We know we can beat anyone on our day, and we will see who it throws up. Wherever we go, we will go there confident because we are coming into a vein of form, and have a few players coming back."
