Richarlison was shown a straight red card by referee Michael Oliver for the 90th minute challenge on Thiago, but Pickford escaped punishment for the fifth minute incident with Van Dijk which saw the defender limp out of the game.
Everton did finish the game down to 10 men as Richarlison was shown a straight red card for another risky lunge, this time on Thiago Alcantara.
The game between the two table-topping Merseyside rivals finished 2-2, with Liverpool having a late Jordan Henderson victor ruled out for a marginal offside call.
Liverpool thought they'd snatched all three points right at the death, only for VAR to deem Mane as offside in the build-up despite the Senegal worldwide looking at least level, if not onside.
Liverpool were humiliated 7-2 by Aston Villa just before the worldwide break but they quickly set about putting things right when Mane scored in the third minute, firing home a low cross from Andy Robertson.
The incident raised further questions about the temperament of Pickford, the England worldwide who has been at fault for some goals conceded by Everton in recent games.
Everton, meanwhile, were top of the table after having won all four of their games in the top flight under Carlo Ancelotti this season. At the time of writing, the score is still level at 1-1.
Van Dijk and Thiago both went to hospital for scans on their respective knee injuries after the game.
Jurgen Klopp's side were left to rue the absence of its main defender just 15 minutes later, as Everton's Keane rose highest in the box to head home James Rodriguez's corner for the equalizer.
But it was Liverpool who remained the more risky side for the remainder of the first-half.
Everton may still be without a victory in a Merseyside derby for a decade - but they will happily take this point against Liverpool after an afternoon of chaos and controversy at Goodison Park. Pickford leapt to his right to make a stunning save from Trent Alexander-Arnold's free-kick before Mane then had a similar chance to the opening goal only to this time skew Robertson's cross wide.
However, the drama did not end there as VAR deemed Mane to be offside as he provided the cross.
Allan and Abdoulaye Doucoure have added energy and a combative edge to a midfield that was one of the worst in recent Everton memory by the end of last season, the Brazilian not quite looking fully fit but still getting around the pitch in that problem-solving way of his.