The Reds were held to a 0-0 draw by the Spanish giants as they bowed out at the quarter-final stage with a 3-1 aggregate defeat but Milner felt the team's display at Anfield was good.
He told BT Sport: "We were obviously the better team. We did enough to win the game.
"The performance, intensity and desire was good. But you can't not take the chances we had and that was the difference.
"The positive is we are creating the chances and the reaction we showed."
Milner believes that last week's first leg loss in Madrid was costly for Liverpool and urged the team to fight for a place in the Premier League's top four.
He added: "We wanted to come out of the blocks and set that tempo. If we had played that first leg with the same intensity it would have been very different. Ultimately we weren't good enough over two games.
"The most important thing is if we perform with the desire and intensity hopefully we can get into those top four spots."
Jurgen Klopp also suggested that it is now crucial for Liverpool to finish in the top four to make sure they return to the Champions League next season.
The Reds boss said: "We really have to keep going, we have to keep fighting, we want to be in the Champions League next year but at the moment we still have to pick up more points than a lot of other teams.
"We love this competition and for other reasons it is very important we are back in it. We can now concentrate on the Premier League and we will. We have to work our socks off against Leeds."