After Conor Gallagher struck for Atleti in the first minute, the Champions League round of 16 second-leg went to penalties with the aggregate score locked at 2-2 after extra-time.
Valverde converted his spotkick in the shootout and after the triumph he conceded they weren't at their best.
He said afterwards, “It was an average display from us at home and here, well, it's always the same, we went out as if we'd already won it. They got their goal in the first move of the match and that hurt us. It's not the same when Atleti are 1-0 up and can defend well, as when they have to come out and attack.
"We handed them a situation where they're very strong. We went to penalties, we were all exhausted, you could really tell. Thank God, we were able to win it."
"For anyone who's ever taken a penalty, I felt such relief when I scored. You're so nervous on the walk to the spot, so much running through your mind. If you miss, you know you have so many people behind you: teammates, everyone who works at the training ground... and if you miss, you're letting them down."
Real will meet Arsenal in the quarterfinals.
Asensio added, “This has to drive us on. There is still a long way to go in the season and this has to be a morale boost so we can keep improving, growing, and keep fighting for everything."