Liverpool went on to claim all three points when Milner made amends from the penalty spot, slamming home after Mane was fouled by Sergio Rico.
Van Dijk, who has been imperious since arriving at Anfield from Southampton in January 2018, looked relieved after the final whistle having got away with a rare mistake.
And the 27-year-old was seen sharing a wry smile with Alisson when the duo embraced at full-time.
Van Dijk also earned a big hug from his manager Jurgen Klopp as the Reds regained top spot in the Premier League.
Virgil van Dijk and Alisson made up after the final whistle against Fulham (Image: GETTY)
Liverpool were 1-0 up at Craven Cottage when the pair were involved in a mix-up (Image: GETTY)
James Milner's sliced clearance was not dealt with by Van Dijk or Alisson (Image: EMPICS)
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Fulham: Ryan Babel’s van Dijk comments mocked by pundits
Speaking after the match, Van Dijk said: "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."
Milner was also sheepish about his role in Fulham’s goal, saying: "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!
“It wasn’t ideal, but luckily enough we got the result."
Both players looked relieved as they celebrated after the final whistle (Image: REUTERS)
Van Dijk, who has been superb this season, was given a hug by manager Jurgen Klopp (Image: REUTERS)