A miscued punt forward from Virgil van Dijk seemed to be heading into the arms of Pickford but, in his attempts to prevent a corner, the keeper spilled the ball underneath his own goal, where Origi was alert to head home.
See 10 of the best snapshots from the Liverpool celebrations below.
Everton - winless at their rivals' ground since 1999 - had a gilt-edged chance to take the lead in the fourth minute when Yerry Mina, incorrectly not flagged for offside, somehow headed Lucas Digne's free-kick wide with the goal at his mercy.
"Against City, you watch it and wonder what they are doing".
"Ali made an outstanding save and then Joe Gomez made an outstanding save".
It was a cruel way to end a polished display from the travelling Toffees.
Klopp ran onto the pitch after the goal
England's number one goalkeeper for some reason tipped a shot that was going over back onto the crossbar, with the ball then bouncing to Origi for an easy tap-in.
Origi, who came on as a substitute for Mohamed Salah in the 75th minute, hadn't scored for Liverpool since May 2017.
What does it mean? . They don't have the same respect for us that they have for Man City, for example.
"It's a lucky, lucky day for Liverpool because they don't deserve the win".
Klopp has come under increasing scrutiny in recent days with Liverpool failing to replicate its devastating attacking form from last season, when it was arguably the most exciting team in Europe.
- Liverpool are unbeaten in each of their last 18 matches against Everton in all competitions (W9 D9) - the Reds' longest unbeaten run against their rivals.
The Premier League title race is still well and truly on.
But prior to that error, especially during the first half, both Alisson and Pickford showed just how important they have both become to their teams.