Van Dijk headed the winner on 92 minutes to earn the 3-2 triumph.
He later said on TNT Sports: "We kept pushing, pushing for the 3-1 and obviously to concede an equaliser, massively deflected, was obviously not great.
"But we found a way again and I think we were dangerous in set-pieces. They were pretty aggressive, I must say. But I'm glad I got the winner.
"(Atletico) a team that never gives up. They have quality players that can punish you if you're not there in a split second, if you are a little bit too late. Obviously they punished us just before the break and after the break we tried to go for the 3-1 but obviously they got a bit of confidence from that as well.
"But eventually you have to keep going and show the resilience that we have been showing throughout many years. It was another great European night here at Anfield."
On Liverpool's spirit, Van Dijk said: "I think we've shown many times that we are capable of doing it. Today I was the lucky one with the winner. We move on.
"It's always good to get off to a winning start. We play at home, it's a good game, it's a good opponent. Another game ticked off, three points."