The Reds forked out a world-record fee for a defender to snare the Dutchman from Southampton at the start of the January transfer window and he marked his debut with the winning goal.
Van Dijk scored a late header to secure a 2-1 win for Liverpool after Gylfi Sigurdsson had cancelled out James Milner’s opener.
Klopp admits Liverpool’s performance left a lot to be desired and was critical of the defending for the Everton goal, but he was happy to see Van Dijk open his account in front of the Kop.
He told the BBC: “It was not the best piece of football for 95 minutes, but a proper fight. I’m delighted with the attitude we showed again – it was a difficult game. It was not the night for brilliant football.
“You would love to see that but Everton defended well. They played football, a big difference to the last game between us.
“The equaliser was the worst protection I ever saw in my life. It made no sense at all. We were all in the wrong direction. But when you make mistakes in football you have to react and we did again.
“It was nice that Virgil van Dijk could score in his first game, a brilliant header. I am really happy. In front of the Kop makes it even better.”