Van Dijk finally sealed his move from Southampton last week, becoming the most expensive defender ever in the process.
Still, very few Liverpool fans seemed too concerned with his price tag in the moments following his victor.
"You can not score a header off a set-piece unless the set-piece is good", Klopp said. His every header was cheered, his every pass judged, his movement analyzed and assessed.
Except for those few seconds, after 84 minutes, when he wandered up for a corner, rose to meet Alex Oxlade-Chamberlain's cross, and ended his first appearance for Liverpool, his first appearance as a record-breaker, with the winning goal in a 2-1 victory against Everton.
Virgil van Dijk enjoyed a dream debut for Liverpool by heading the victor in front of the Kop in the 230th Merseyside derby as they knocked Everton out of the FA Cup. "To score a goal is even more special".
However, referee Bobby Madley, despite being in the thick of it, chose to take no action against either player.
Holgate's head was scrambled and five minutes later, after losing possession, he shoved Firmino into the front row of the Main Stand.
The hosts had to defend doggedly to shut down a promising attack for Everton in the 24th minute, with Van Dijk playing his part to thwart them before Milner side-footed a volley wide of the far post at the other end.
Liverpool and Everton duelled out a lively Merseyside derby on a night where tempers flared and goals were scored.
The morning's headlines will be about Van Dijk, not about how much he cost, or how much of a risk he represents, but about the most ideal introduction he could have asked for.