Jurgen Klopp's side took an early lead through Adam Lallana's stylish finish, but weaknesses at the back allowed Dimitri Payet's free-kick.
Jurgen Klopp's side took an early lead through Adam Lallana's stylish finish, but weaknesses at the back allowed Dimitri Payet's free-kick and a neat finish by Michail Antonio to give West Ham a surprise half-time lead at Anfield.
Loris Karius, the Liverpool goalkeeper, was arguably at fault for Payet's goal, but then saw Hammers counterpart Darren Randolph make a mistake for Origi's equaliser by failing to hold Sadio Mane's cross.
The 2-2 draw did at least lift West Ham out of the relegation zone, although it extended their run without a Premier League victory to six matches.
It looked like being a gloomy day for Slaven Bilic's side when they conceded a well-worked goal in the first five minutes, as Mane crossed from the left and Lallana took a touch to control before steering a shot past Randolph.
Yet even in the early stages Liverpool's defensive frailties that marked last weekend?s 4-3 defeat at Bournemouth were apparent once again.
Antonio found space too easily as he sprinted between centre-backs Dejan Lovren and Joel Matip to collect Winston Reid's long ball over the top, but Karius turned aside the shot at his near post.
Karius was then caught out as West Ham drew level in the 27th minute, getting a hand to Paye's free-kick as it was floated over the wall, but failing to stop it.
That equaliser affected Liverpool's rhythm, despite half-chances for Jordan Henderson and Roberto Firmino soon afterwards, and they went behind six minutes before half time to another soft goal.
This time Havard Nordtveit's long diagonal pass out of defence flicked off Henderson to wrongfoot Matip, who was caught on his heels as Antonio ran into the space behind him to flick the ball beyond Karius -- who was caught in no-man's land -- and in.
Liverpool could have been level by the interval but Firmino glanced wide from Georginio Wijnaldum's cross and then Lallana had a penalty appeal rejected after tumbling with Payet's arm across his chest.
Referee Mark Clattenburg gave only a corner and from it Matip nodded Henderson's delivery on to the face of the crossbar.
Klopp's team were though level within three minutes of the restart. Mane spun to cross from the left, Randolph lost his balance and the ball as he stretched to try to gather, and Origi turned it in via the post.
Spirits restored, Liverpool almost regained the lead within moments, as Wijnaldum steered a shot just wide on the turn from the edge of the penalty area.
Wijnaldum had another good chance just after the hour mark after being played into space down the left, but placed his effort beyond the far post.
Henderson then bent a fine 25-yard shot towards the top corner midway through the second half, only for Randolph to deny him with magnificent flying save.
Bilic brought on former Liverpool striker Andy Carroll with just under half-an-hour to play.
The home side finished the stronger, though, with full-back Nathaniel Clyne blazing over after Lallana's clever backheeled flick from Mane's low cross.