Two group-stage exits were followed up with defeat against Barcelona in the last 16 last term, and Milner is eyeing a big improvement in this year's competition.
He told reporters: "There are a lot of games we've played now, a lot that we have learned over the years we've been in it.
"I think having won the Premier League a couple of times and been fortunate enough to win every trophy domestically, we want to prove ourselves on the European stage and I believe we have a good enough team to do that.
"I think being the club that we are, and with the players we have, winning it is the ultimate aim. But you never get too far ahead – you look at it a game at a time and start then."
The champions travel to Bayern Munich on Wednesday.