Everton looked to have done enough to secure a point but a horrendous error from goalkeeper Jordan Pickford gifted the ball to Origi, who headed it home in the sixth minute of stoppage time.
Chasing a first win at Anfield since 1999, Everton created the better of the openings in the first half, but found Liverpool goalkeeper Allison in inspired form.
Liverpool upped the tempo after the break and twice went close through Mohamed Salah and Sadio Mane.
Origi, who came off the bench for his first league appearance of the season, saw an earlier close-range effort come out off the post, before he grabbed the headlines with the bizarre winner.
The goal sparked wild celebrations at Anfield with manager Juergen Klopp running on to the pitch. The win allowed second-placed Liverpool to close the gap on league leaders Manchester City to two points.
(Reporting by Peter Hall; editing by Clare Fallon)
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