First half goals from Darwin Nunez and Cody Gakpo coupled with goalkeeper Nick Pope's dismissal were all Liverpool needed to wrap up the win in an eventful opening period.
Both Newcastle defeats in the top flight this season have come against Liverpool and Van Dijk, who returned from injury to martial the back for the Reds, reckons the win can help Jurgen Klopp's side climb up the table in what has been a disjointed season so far.
Asked if it was a significant result for the Reds, Van Dijk told Sky Sports: "Let's hope so.
"We have had results in the season already where we felt like now is the time to kick on, but this feels a little bit different.
"I think coming away from the big win against Everton, it was important for us to keep going, find a way to win, show your intensity, particularly in the beginning.
"The red card changed the game a little bit for them, also for us, and we could've done better.
"But to win here…there's a reason we are the only team so far this season that has beaten them [in the Premier League], so I'm very proud of the boys."
Liverpool's win took them to 8th position, six points off the top four.
