Everton looked to have done enough to earn a point from an admirable performance in which they created the better chances when Liverpool were awarded a free-kick in the 96th minute.
Alisson played it to Trent Alexander-Arnold and his long punt was cleared only as far as Virgil van Dijk, whose skewed long-range effort looked to be going out of play.
However, Pickford inexplicably palmed it into the air and onto the crossbar, the ball dropping into the path of Divock Origi, who nodded into an empty net and snatched the win for the Reds, keeping them within two points of leaders Manchester City.
Pickford understandably accepted blame and, speaking to Sky Sports, he said: "[It's] a little error from me, a bit of a freak one really. I think it's hit the bar twice, I've tried to flick it over, it's spinned back and as I've went to flick it over I think my hand's hit the bar as I've touched the ball. It's unfortunate but I'm strong mentally and I'll get ready for Wednesday [against Newcastle United].
"I think that's the Everton luck when we come to Anfield, there's nothing I can do [but] hold my hands up. I'm strong mentally and it won't affect us for the next game, just got to get on with it and say sorry to the Everton fans because I think you can see our progression as a group, the chances, the football we've played today, I think that's always a positive. It's my fault at the end of the day.
"We held our own so we take positives from it, freak goals happen sometimes and it's always the case as a goalkeeper if you make a mistake, you make an error, it's always going to bite you. Unfortunately it's been me today but move on and I'll go and perform on Wednesday to my best.
"We know we held our own against a good Liverpool side and we know it's always going to be tough coming to places like this. It's just unfortunate but we go on.
"It won't affect my performances in the future, I've got to move on, it's one of them things that will happen now and then, it's how you bounce back and your mental strength will get you through it."
Manager Marco Silva had consoling words for Pickford, and insisted he had no problem with his Liverpool counterpart Jurgen Klopp running onto the pitch in celebration after the winner.
Asked what he told the players in the dressing room, Silva told Sky Sports: "That we didn't deserve it, that I was proud of them. Like we started the game altogether, we finish the game altogether. The draw would have been a fair result.
"We will analyse the moment with Jordan, it was a bad moment for him. But I told him he has done many amazing things. It was a lucky day for Liverpool.
"I didn't see Klopp's reaction and if we scored that goal and had been so lucky I might have done the same. I didn't see any disrespect. Their fans might have expected another easy match but their celebrations at the end showed they got something different."