Richarlison was sent off late on for a terrible challenge on Thiago but the hosts held on for a draw.
Mane returned after returned his positive COVID-19 test to open the scoring inside four minutes with a fine finish from Andy Robertson's cross.
Three minutes later, Liverpool wanted a penalty when Van Dijk was on the end of a reckless challenge from Pickford but he was offside and the goalkeeper didn't even receive a yellow card.
Van Dijk limped off and Everton equalised after 19 minutes when Keane converted from Lucas Digne's cross.
Salah put the visitors ahead after 72 minutes with an instinctive wonder strike inside the area to score his 100th goal for the club.
Joel Matip almost scored a third from a corner but Pickford produced a stunning save to keep it out. It turned out to be an important stop because Everton made it 2-2 with a fantastic header from Calvert Lewin.
The England international becomes the first Everton player to score in the first five league games since Tommy Lawton in 1938-39.
Henderson scored late on but the goal was ruled out by VAR.
"I don't know where the line is where you can do offside," Klopp told BBC when asked about the late disallowed goal.
"Yes, we should have won this game. The boys played a super game. against a side with quality and confidence. Dominating from the first second is absolutely exceptional."
After the match, Richarlison released a statement on social media to apologise for his challenge on Thiago.
"Anyone who knows my background knows that I've never been a violent player," he wrote. "What happened today was a fatality due to sheer excess of dedication. I did not challenge that play with the intention of hurting Thiago [Alcantara].
"When I realized we wew going to crash, I retracted my foot but my knee caught him on the way thorugh, which was inevitable due to the speed of the play. I've already sent a message to him apologizing and I am also doing it publicly.
"I hope he is fine. I also apologize to my teammates, the club and the fans for leaving them with 10 men. Unfortunately, there is nothing I can do to change what happened, other than to reflect, work and move on."