The Reds grabbed a 2-0 victory at San Siro in the first leg of the Champions League last-16 tie.
Van Dijk admitted his team suffered during the clash but ultimately deserved their victory.
"It's a tough place to come, a really good team, a good system," van Dijk said. "They were difficult to break down. The atmosphere was incredible, I really enjoyed it out there.
"Obviously it was very important with the set pieces, it's a strength of ours. Hopefully we can keep using this. Overall it's a very solid performance and a good result to take back home. I think it's a well-deserved win, but it was very hard work.
"Everyone expected to be under pressure, because they are a very good team and it is the Champions League. We needed to be ready to suffer and do the hard work, and we definitely did. Happy days."