The Dutchman, signed last week from Southampton for a reported fee of 75 million pounds ($101.7 million), rose inside the area to head home an Alex Oxlade-Chamberlain corner in front of the Kop end in the 84th minute of what had been an enthralling Merseyside derby.
"What a night. It was thoroughly deserved, it felt amazing to be out there, the goal is very special for me and my family," said Van Dijk.
"Playing at Anfield for Liverpool is a dream for every player. To score a goal is even more special."
Liverpool looked to have been frustrated after Gylfi Sigurdsson brought Everton level in the 67th minute after a brilliant counter-attack from Sam Allardyce's side.
That looked to have produced a repeat of the Premier League clash between the pair in December when Everton grabbed a 1-1 draw despite their rivals having had many more chances.
But Van Dijk, bought to shore up Liverpool's weakest link, the centre of their defence, proved his value at the other end of the field with his strength and aerial power delivering the killer blow.
After a frenetic start, Liverpool had taken the lead through a 35th minute penalty from James Milner after Everton defender Mason Holgate had pulled down Adam Lallana in the box.
It was in many ways an old-school derby with plenty of strong challenges, a high-tempo and end-to-end rhythm but it was Juergen Klopp's side who created the greater openings.
Everton's Wayne Rooney, who had been booked in just the seventh minute for a late lunge on Joe Gomez, was substituted in the 52nd minute, probably to save him from the risk of a red-card as he looked close to loosing his cool.
Gomez headed wide at the back post when he had been left unmarked at a 54th minute corner and a minute later Lallana poked the ball wide after being sent clear through on goal.
Everton keeper Jordan Pickford produced a fine parry to keep out an Andy Robertson blast at the near post.
The Toffees looked to have made Klopp's men pay for failing to make the most of those chances with Sigurdsson's cool side foot finish.
But the world's most expensive defender had other ideas.
($1 = 0.7373 pounds)
(Reporting by Simon Evans; Editing by Christian Radnedge)
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