Liverpool were comfortable 3-0 winners on Sunday.
Van Dijk later said: "It's a proud moment. What pleased me the most was the performance from start to finish. We want to enjoy ourselves out there and the only way to do that is with the hard work we put in. We dominated and should have scored more. Winning 3-0 is a big improvement from last year.
"We had to stay focused and we did very well. I'm very proud of the boys.
"Joshua Zirkzee plays now as a striker for them but I know he likes to drop and receive the ball to his feet. We know they like to do this passing in behind but we nullified it quickly. Our game plan worked. We kept patient.
"What Jurgen Klopp said in one of his last interviews is the new manager inherited a very good squad. The manager left but the players didn't. We are not satisfied with just competing.
"We want to achieve greatness this season. I'm not competing in any competition just to compete, I want to win. We come up against a very good team in Man City and Arsenal. Our consistency is looking good, after the international break could be interesting when playing every three days. I'm certainly enjoying the ride."