Liverpool were 1-0 up through a first half strike from Sadio Mane when Van Dijk was guilty of selling goalkeeper Alisson short with a back header that allowed former Reds winger Ryan Babel to steal the ball and equalise for Fulham.
It then took a much-debated penalty, converted by Milner, to win the game and send Liverpool back to the top of the Premier League table.
"Mistakes happen in football. It is what it is, but we go again. I'm not giving excuses, they scored, but we scored again and we go home top of the league," Van Dijk told Sky Sports soon after the final whistle, reflecting on his error.
But Milner intervened and pulled blame over to himself after it was his botched clearance - sending the ball up in the air and towards his own goal - that had given the Liverpool centre-back the problem to deal with in the first place.
"I'm not giving excuses.... " - Virgil van Dijk— Optus Sport (@OptusSport) March 18, 2019
"It was me who put him in the mire to be fair... " - James Milner
James Milner chuckles at the quagmire he created for Liverpool teammates Virgil van Dijk and Alisson Becker.#OptusSport #FULLIV pic.twitter.com/0HMFJcRyOk
"I feel the goal was probably my fault. It was me who put him in the mire," he said.
"The gaffer said come on and calm it down so the best thing to do is slice it over the top and put Virg under pressure, which wasn't ideal, but luckily enough we got the result."
Remarkably, Milner has still never lost a Premier League game in which he has scored. This was the 51st Premier League goal of his top flight career, stretching back nearly 17 years to 2002 when he netted his first for boyhood club Leeds aged just 16.
Liverpool will remain top of the league while domestic football pauses for the international fixtures. They will then hope to hold onto their lead and pile the pressure on Manchester City when they face Tottenham at Anfield in a huge top-three clash at the end of the month.