The striker hit home in the 94th minute to break open a game that appeared to be heading towards stalemate, sending the Reds back up to second place behind Chelsea.
Sadio Mane’s dramatic injury-time winner helped Liverpool keep pace in the Premier League title race and earn Merseyside derby bragging rights in a 1-0 win at Everton on Monday.
There had been little excitement during a largely forgettable encounter at Goodison Park until Mane pounced on the rebound after Daniel Sturridge’s shot struck the post in the 94th minute to earn a big three points for the Reds.
Neither side produced much in the final third during a drab first half and, although Everton edged proceedings, there were no shots on target in the opening 45 minutes – only the second time that has happened in the Premier League this season.
Liverpool improved after the break and Roberto Firmino should have done better with the game’s first clear-cut chance that was saved by Maarten Stekelenburg, while the Brazilian saw a volley brilliantly saved by Everton’s substitute goalkeeper Joel Robles.
But Mane’s late intervention rendered those misses irrelevant and Liverpool are now unbeaten in 12 games against Everton, while the gap to in-form leaders Chelsea remains at six points ahead of the Christmas period.
A limp Ashley Williams header was all Everton – who were fortunate not to have Ross Barkley sent off for a lunge on Jordan Henderson – could muster and they remain in ninth place.
Everton set the early tempo in a frenetic opening that produced little quality in the final third, the hosts full of energy as they looked to hurry their neighbours.
It seemed to do the trick as the visitors struggled for any sort of fluency and Klopp was visibly frustrated as he barked orders from the sideline to no avail.
As the half wore on Divock Origi was finally presented with a sight of Stekelenburg’s goal but he blazed Nathaniel Clyne’s cross high and wide.
The Toffees were no better with Enner Valencia, Romelu Lukaku and Ross Barkley all failing to test Simon Mignolet, while Ramiro Funes Mori headed the clearest opening for either side wide from the half’s only corner two minutes before the break.
Liverpool had a great chance five minutes after the restart when Firmino shrugged off Funes Mori on the edge of the penalty area, but his attempted lob was well saved by Stekelenburg and in the chaos that ensued Mane and Clyne both failed to convert.
Everton lost their Dutch goalkeeper to injury just after the hour and they were lucky not to see Barkley sent off after 68 minutes for a mistimed lunge that saw him catch Henderson flush on the ankle.
With chances at a premium Williams should have done better with a free header after 75 minutes – Everton’s first on target of the match.
Firmino was denied by a superb reflex save from substitute keeper Robles as the pressure on Everton’s goal began to grow and in stoppage time their resolve was broken.
Substitute Sturridge held off the challenge of Leighton Baines before firing low past Robles, the ball hitting the upright and rebounding to Mane for a simple tap in.
Firmino almost added a second, but one goal was enough to see Liverpool back into second place.