The Reds took a 1-0 lead in the first half when James Milner converted from the spot on Friday, reports Efe.
Everton were more aggressive in the second period, equalising in the 67th minute with a goal by Gylfi Sigurdsson.
With six minutes left in regulation at Anfield, Van Dijk headed-in Alex Oxlade-Chamberlain's corner to give Liverpool the victory.
To score the winning goal in the Merseyside derby was an auspicious debut for Van Dijk, who joined Liverpool from Southampton for ?75 million ($100 million), a record transfer fee for a defender.
Just up the road at Old Trafford, United struggled more than expected against second-division Derby, who benefited from a solid performance by goalkeeper Scott Carson.
The English Premier League (EPL) side didn't open the scoring until the 83rd minute, when Jesse Lingard scored with a magnificent effort from distance.
Lukaku, who came on in the second half as a sub, sealed the triumph with a goal in the final minute of regulation.
(This story has not been edited by Social News XYZ staff and is auto-generated from a syndicated feed.)
