Naby Keita played Roberto Firmino through on goal late on but was unable to find the target, and substitute Origi hit the bar from a corner as Liverpool continued to push for a victor.
The Blues put on one of the best away derby performances of recent years, but it wasn't enough to counter the strangest goal seen in a Merseyside derby since Pepe Reina dropped Lee Carsley's shot onto Andy Johnson's head in the Goodison derby a decade ago.
Reds manager Jurgen Klopp - who later admitted that he had thought Virgil Van Dijk's punt up field virtually signalled a concession of a last chance for a win - immediately responded by running wildly onto the pitch, celebrating passionately with his own goalkeeper Alisson, who had instigated the free-kick from where the vital goal would eventually be the result.
Alisson and Jordan Pickford have both solved that problem for Liverpool and Everton respectively, but Pickford suffered the cruel blow that all keepers are susceptible to by conceding a stoppage-time goal after an individual mistake.
The two goalkeepers exchanged crucial saves, with Alisson Becker spreading himself to block a header from Andre Gomes, who had seemed destined to score from four yards out when Theo Walcott directed Bernard's cross perfectly into his path.
Pickford also made his mark with a crucial save on Xherdan Shaqiri in the 34th minute.
Arsenal's Pierre Emerick Aubameyang celebrates his goal in the 4-2 win over Tottenham Hotspur at the Emirates Stadium in London
Richarlison then made his way past the Liverpool defence with a weaving run, but stumbled as he was about to shoot and the chance went begging for Everton.
Everton made six new signings in the summer, as well as appointing a new manager in Marco Silva, and the freshness was evident against Liverpool until the late, cruel victor conceded by the home side.
Prior to this game, Liverpool had avoided defeat in their last 15 league meetings with their neighbours, with Everton famously not having won at Anfield in any competition since 1999.
Despite their woeful record at Anfield Everton played with plenty of bravado and it produced an end-to-end spectacle reminiscent of the blood-and-thunder derbies of yesteryear.
Mina was commanding at centre-half and Digne impressive down the left flank.
Everton's chances were just as good. Both teams tried really to play football.
Un temblor sacudió parte del Conurbano y de la Ciudad
Minutos después, desde el Instituto Nacional de Prevención Sísmica ( INPRES ) confirmaron que se trató de un sismo . Esta mañana se registró un sismo durante el inicio de la Cumbre del G20 en Buenos Aires , Argentina .