Goals from Ryan Gravenberch and Hugo Ekitike were Liverpool’s reward for a dominant first half and enough for a fifth consecutive Premier League win.
But Slot’s men ran out of steam after the break in their third game in seven days and were left hanging on after Idrissa Gana Gueye fired Everton back into the contest.
Liverpool had needed late goals to win all of their previous four Premier League games and Champions League opener against Atletico Madrid in midweek.
This time the Reds did the bulk of their good work inside the first 30 minutes to leave Everton still without a win at Anfield in front of a crowd in the 21st century.
“We needed mentality today in a different fashion than the few times before,” said Slot.
“You can see how well we can play when we are fresh in the first 45 minutes. We ran out a bit of energy but we didn’t run out of mentality.”
Slot left both his big money summer signings Florian Wirtz and Alexander Isak on the bench, but Liverpool were better balanced thanks to the return of Alexis MacAllister alongside Dominik Szoboszlai and the impressive Gravenberch in midfield.
The Dutch midfielder broke the deadlock just nine minutes in with a deft finish on the half-volley from Mohamed Salah’s cross.
Everton had been on a four-game unbeaten run thanks in large part to the form of the resurgent Jack Grealish.
Nervy finale
The Manchester City loanee was unsurprisingly at the heart of their best first-half move with a pass through to Kieran Dewsbury-Hall who fired wide from a narrow angle.
Liverpool responded in style to a fine team move culminated in Ekitike rolling Gravenberch’s pass through the legs of Jordan Pickford for his third goal in five Premier League games since his move from Eintracht Frankfurt.
But the second half was a completely different story as Liverpool almost saw another two-goal lead disappear after being pegged by Bournemouth, Newcastle and Atletico already this season.
Fatigue this time seemed to play a part with kick-off just over 60 hours after Virgil van Dijk’s stoppage time winner against Atletico on Wednesday.
“We lost the game in the first half. Frustrating because if we had played like that, pressed like that from the start it would have been a different game,” said Grealish.
Everton had a lifeline just before the hour mark when Grealish’s deep cross was turned back into the path of Gueye, who blasted his past Alisson Becker.
Slot turned to Wirtz and Isak off the bench to try and wrestle back control to little effect.
Liverpool, though, did manage to see out a nervy finale to inflict an unwanted record on Everton boss David Moyes.
The Scot has now failed to win in 23 trips to Anfield as a visiting coach, the most any Premier League manager has played at a stadium without victory.
Arsenal late show
In yesterday’s late game Gabriel Martinelli’s 93rd minute equaliser saved Arsenal’s Premier League title bid from a damaging defeat to Manchester City.
The Gunners were heading to a second defeat in five Premier League games due to Erling Haaland’s early strike on the counter-attack for City.
Martinelli came off the bench to score for the second time in a week to salvage a 1-1 draw, but both sides lost more ground to Premier League leaders Liverpool.
Arsenal went second, five points behind the champions, while City are eight points off the top in ninth.
Mikel Arteta’s men had not conceded from open play in their first five Premier League and Champions League games this season, but were opened up on nine minutes by the scintillating form of arguably the world’s best striker.
Haaland started the move deep inside his own half with a lay-off to Tijjani Reijnders and then sprinted upfield to receive the Dutch international’s pass before coolly slotting beyond David Raya.
The Norwegian already has 13 goals for club and country in just eight games this season.
Guardiola sacrificed Haaland and Phil Foden in the second half to bolster his back line but the gamble backfired when Martinelli galloped onto Eze’s ball over the top and looped a fantastic finish over Donnarumma.
In yesterday’s early games Sunderland and Aston Villa drew 1-1, as did Newcastle and Bournemouth.
Beleaguered
Meanwhile, Manchester United beat Chelsea 2-1 in a chaotic match at Old Trafford on Saturday to earn vital breathing space for beleaguered manager Ruben Amorim.
All eyes were on a rain-lashed Old Trafford for the early evening kick-off, with Amorim in desperate need of a win after a terrible start to the season.
The home side were given a huge helping hand when Chelsea goalkeeper Robert Sanchez was sent off in the fifth minute of a stormy encounter.
Chelsea boss Enzo Maresca made frantic personnel changes to compensate for the loss of Sanchez but the match immediately became an exercise in attack versus defence.
United captain Bruno Fernandes made the breakthrough in the 14th minute, prodding home from close range in his 200th Premier League game.
Casemiro doubled the lead before the Brazilian midfielder was shown a second yellow card on the stroke of half-time for tugging back Andrey Santos.
Even though both tems were now down to 10 men, Chelsea struggled to create chances in sodden conditions until Trevoh Chalobah brought the visitors back into the match with a bullet header 10 minutes from time.
But United survived some nervy moments to record just their second win of the season, lifting them into the top half of the table. It was a first league defeat for Chelsea.
Bleak
Graham Potter’s future as West Ham boss appears bleak after a fourth defeat in five matches left the club firmly rooted in the relegation zone.
Hammers boss Potter admitted that his side’s 2-1 loss to Crystal Palace was a “tough moment” as he fights to keep his job.
Former Nottingham Forest boss Nuno Espirito Santo and ex-Wolves manager Gary O’Neil have been linked with the position at the London Stadium.
Asked whether he felt he still had the support of the Hammers’ hierarchy, he replied: “I’ve no reason to think not.
“But I also understand the environment and the results. But I have no complaints about the support I have had.”
Tottenham came from two goals down to draw 2-2 with Brighton. Goals from Yankuba Minteh and Yasin Ayari gave the home side a cushion before Richarlison pulled one back shortly before half-time and Jan Paul van Hecke’s own goal made it all square.
Newly promoted Leeds recovered from conceding the first goal to beat Wolves 3-1, leaving the Midlands club without a single point from their five matches.
Burnley and Nottingham Forest shared the spoils in a 1-1 draw.