Liverpool s Divock Origi, left, celebrates with Liverpool s Mohamed Salah after scoring his sides second goal during the English Premier League soccer match between Liverpool and Everton at Anfield stadium in Liverpool, England, Sunday, April 24, 2022 (AP)
The Toffees' day got off a terrible start even before a ball was kicked at Anfield as Burnley's 1-0 win over Wolves dumped Everton into the relegation zone.
Second-half goals from Andy Robertson and Divock Origi ensured they remained there despite a spirited display in a feisty encounter.
A 12th consecutive home league win for Liverpool cuts City's lead at the top of the table back to one point with five games remaining.
Everton have still not won away from home in the league since August, but they dispelled criticism of being a soft touch away from Goodison Park by showing the fight they will need in the remaining six games to maintain their 68-year stay in the top-flight.
In stark contrast to Manchester United's capitulation in losing 4-0 at Anfield in midweek, Frank Lampard's men at least managed to make the Reds earn the three points.
A Sadio Mane dipping effort from the edge of the box was the sum total of the threat to the Everton goal in a scrappy first 45 minutes littered with stoppages in play.
Mane was lucky just to see yellow from one melee involving almost all 22 players after Liverpool refused to put the ball out of play with Richarlison down injured.
Everton's 32 completed fewest passes in the first half was the fewest of any team in a Premier League game since 2006.
However, they posed the greater threat early in the second half and were made to rue Anthony Gordon's chance to open the scoring when he dragged his shot across goal with just Alisson Becker to beat.
Gordon also had claims for a penalty appeal waved away to Lampard's dismay for a push by Joel Matip.
With the scores still level on the hour mark, Klopp was forced to turn to his bench with Luis Diaz and Origi, a perennial scourge of Everton in derbies past, thrown on.
The all-important goal, though, came from an unlikely source as Robertson headed in just his second goal of the season from Mohamed Salah's dinked cross to the back post.
Salah sliced another good chance over as Everton wobbled after going behind.
But the visitors found their feet again to ensure a nervy finale for the vast majority of the crowd inside Anfield.
Demarai Gray fired a powerful shot inches wide before Robertson was also a hero at the other end as he cleared Dele Alli's dangerous cross to prevent Alex Iwobi applying a finishing touch.
Origi looks set to depart Liverpool at the end of the season after falling further down the pecking order since the January arrival of Diaz.
But the Belgian signed off with his sixth goal in 12 Merseyside derbies when he headed in at the far post five minutes from time.
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