An even Merseyside derby was heading towards a goalless draw when, in the sixth minute of stoppage time, Origi scored his first goal this season as the ball fell to him after Jordan Pickford had touched Virgil van Dijk's skied shot on to the bar.
In an enthralling first half, Everton's Yerry Mina headed wide from seven yards early on.
Alisson then saved Andre Gomes's header from point-blank range, before Joe Gomez cleared the rebound off the line.
At the other end, Pickford did well to deny Xherdan Shaqiri.
Liverpool raised the pressure in the second half and Salah and Mane each shot wide.
Shortly after hitting the bar from close range, Origi made no mistake with his second chance.
The win keeps Liverpool second, two points behind Manchester City, while Everton stay sixth on 22 points.
Liverpool: 5 Dec v Burnley (A)
Everton: 5 Dec v Newcastle United (H)