The Reds went down 3-1 in the initial meeting in the quarter-final tie, though Mohamed Salah's away goal in the Spanish capital provided hope of a comeback.
However, the Egyptian missed a glorious early opportunity to set the tone for Jurgen Klopp's side on Wednesday, with Liverpool only managing to get four of their 15 attempts on target against a Madrid defence once again without Sergio Ramos, Raphael Varane and Dani Carvajal.
Zinedine Zidane had to use midfielder Federico Valverde as a makeshift right-back, yet the reigning LaLiga champions held on for a 0-0 draw that sets up a last-four tie against Chelsea.
"I thought the performance was good, the intensity and desire was good," Milner told BT Sport.
"We won the ball back well, pressed really well. But you can't not take the chances we had tonight. We had more than enough chances to do what we needed to do and we didn't take them, unfortunately.
"The positives are that we are creating them, especially against a team like this after the first leg, and we have done enough to go through. I believe that with the chances we had.
"We were obviously the better team. Alisson has made a good save as well [from Vinicius Junior], which you expect. But we did enough to win the game.
"You only need that first one – that first one changes the game. But, if you don't take your chances, then you're not going to go through.
"It's a disappointing way to end, because if we had played that first leg with the same intensity it could have been a different story.
"You can say we've left ourselves with too much to do because we haven't done it, but we have done enough if we take our chances. Ultimately, we weren't good enough over the two games."
For just the second time in Klopp's reign, Liverpool failed to win either leg of a Champions League knockout tie. The other occasion came in the 2019-20 season when Atletico Madrid knocked out the Premier League side in the last 16.
Milner - who became the oldest player to start in a European Cup/UEFA Champions League knockout game for Liverpool since Ian Callaghan in March 1978 - hopes the squad can build on the performance, even if the result was not what they wanted.
Liverpool must now fight to qualify for next year's competition through the domestic route, meaning they must secure a top-four finish in the league. The Reds currently sit sixth.
"If we perform at that intensity, that desire and tempo that we showed tonight over the last league games, then hopefully we can get into those top four spots," the midfielder said.
"That's the most important thing – that we replicate this form and these sorts of performances. That's the one thing we’ve been missing, that consistency.
"We need to get that back and push these last few games, put this disappointment to one side."