The game seemed to head for a draw before Origi, brought on in the 84th minute, nodded in from close range after the ball hit the cross bar and bounced back.
The win means Liverpool remains just two points adrift of Premier League leaders Manchester City after 14 games.
Liverpool has piled the pressure on Everton all game, but were unable to break the resolute defence marshaled by Yerry Mina and Michael Keane.
The Reds hit the woodwork on three occasions as they tried to force a lead, but had to wait until stoppage time to get their reward.
With the game winding down, Liverpool manager Jurgen Klopp rang the changes, bringing on Naby Keita, Daniel Sturridge and Origi in place of Xherdan Shaqiri, Mohamed Salah and Roberto Firmino respectively.
Origi was appearing for Liverpool for the first time this season, having fallen behind the pecking order at Anfield.
His goal may give Klopp something to consider as the Premier League heads into a busy December schedule.
Liverpool’s next game will be on Wednesday against Burnley
