The Reds ran out 3-1 winners at Anfield and were comfortable throughout the game.
And Van Dijk says they were determined to ‘make it right’ after going into the game without a Premier League win against them since March 2014.
“It was special,” the 27-year-old said.
“It was a long time ago that we won against them at home.
“We wanted to make it right and we wanted to show everyone that we’re doing pretty well.
“We had to be brave. We knew what we were going to face.
“They obviously have a lot of quality, especially up front, but we did well – we didn’t give them any chance to have clear options. It was a deserved win.”
Sadio Mane put the hosts in front with a calm finish from Fabinho’s superb pass on 24 minutes.
Jesse Lingard gave the visitors some brief hope when he leveled just after the half-hour mark but the introduction of Xherdan Shaqiri turned the game on its head.
The Swiss star replaced Naby Keita in the 70th minute and restored the lead just three minutes later with his first of two deflected efforts before sealing the three points with 10 minutes remaining.
Liverpool went back to the top of the Premier League table and are still unbeaten after 17 games.
And Van Dijk says his side showed great character to recover from the disappointment of Lingard’s equaliser.
“You can concede and it happened,” he added.
“The thing then was to bounce back and make sure we kept pushing, putting the pressure on them and being patient.
“That’s what we did and in the end, big Shaq came on and did the job for us.
“He is a fantastic player. He showed it since the first game against Manchester United in America [in pre-season].
“He is something different and sometimes that’s a game-changer.”