The Reds looked dangerous in Slot's first game at Anfield as goals from Luis Diaz and Mohamed Salah secured a 2-0 win and offered optimism that the Dutchman can replicate the success of the German.
Van Dijk said Liverpool enjoyed an "amazing" spell under Klopp - who helped the club win their first Premier League title in 30 years in 2020, as well as lifting the Champions League, the FA Cup and the Carabao Cup twice.
The centre-back said the team could not "forget" what they had achieved under Klopp but claimed it was important to focus on working hard for Slot and create even more memories under his leadership.
"The time with Jurgen is amazing and the time with the new boss is a joy and he has a certain way of playing," he told Sky Sports. "We work a lot on so many good things and you see a lot with and without the ball that gives confidence to the team.
"We have to, I won't say forget about the Jurgen period, but that is in the past and we had an amazing time and now it is a time for the new boss to hopefully be successful. That's why we have to focus on that, not have comparisons and all of this stuff in my opinion.
"We don't have new players so far but obviously [I have] that extra responsibility because I know the players inside and out. But I am here for him [Slot]. I told him that from day one and I will help them and the backroom staff. I want to be successful and we are in the same boat. There is extra responsibility there."
Van Dijk hailed the "amazing" work done by Jurgen Klopp at Anfield
Van Dijk said it was important to focus on creating memories under Arne Slot
The 2-0 win came after a victory of the same scoreline against Ipswich Town on the opening day, with Liverpool earning admirers for their attractive style of play under the Dutchman.
The team will face a stern test of how far they have come under Slot next, however, with the Reds set to travel to Old Trafford to face rivals Man United next week.
Van Dijk also spoke about his future during his interview, with the Dutchman's contract at Anfield expiring at the end of the season. He said he was "very calm" on his future, adding: "The thing is, I want to play the best season I can again. I want to be important and stay important for the club this year. And whatever happens next year, we'll see.
"Everything happens for a reason, that's how I see it. And for now there have been no changes in my situation, but I'm very calm about it - and there's no reason for me to start thinking about something else because I have a whole season to play for still. Hopefully it'll be successful."
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