Jürgen Klopp made five changes for the clash – one enforced – making clear his priority is next week's Champions League quarter-final second leg at Manchester City.
Everton finally threw caution to the wind with 15 minutes remaining but it was too little too late and their wait for victory against their great rivals rumbles on.
But what did we learn from the game?
Pickford enhances England No.1 claim
The Toffees stopper got the nod in one of England's two friendlies last month, the 1-0 win over the Netherlands, and Three Lions coach Gareth Southgate was at Goodison Park at lunchtime to take in the Merseyside derby.
And Pickford only enhanced his chances of making the No.1 shirt his for the World Cup in Russia.
In the first half alone he made two excellent saves, first denying Dominic Solanke from point-blank range with a reaction stop, then palming out James Milner‘s bending effort towards the far corner. Although he had more time to make the latter it was more impressive in that he had time to think, rather than reacting purely on instinct.
There are no question marks against Pickford's distribution or shot-stopping; only his decision-making and that, too, was flawless.
Clyne makes impressive return
The impressive form of young right-backs Joe Gomez and, particularly, Trent Alexander-Arnold has been one of the stories of the season for Liverpool. The pair have stepped up admirably in the absence of Nathaniel Clyne to the point there has been no question of Klopp bringing in a more senior alternative.
An unused substitute in last weekend's win over Crystal Palace and the Champions League demolition of Manchester City, Clyne got the nod this afternoon for the first time since the final day of last season.
Plagued by a back injury since the summer, the England international has hardly been missed, so impressive have Gomez and Alexander-Arnold been.
But against the Toffees it was as if Clyne had never been away. The 27-year-old attacked Leighton Baines with relish and 51 per cent of Liverpool's attacking play was directed down his side.
He ought to have marked his comeback with an assist and would have were it not for Pickford's fine reaction stop to deny Solanke early on. Defensively he was sound, too, keeping Yannick Bolasie quiet by making two tackles, one interception and two clearances.
Toffees fail to trouble Klavan
Even with a 3-0 lead in hand ahead the second leg of their Champions League quarter-final, the timing of this game was not ideal for Liverpool.
That meant promoting Ragnar Klavan to the starting XI in an unfamiliar left-back role while Under-18s winger Rafa Camacho was named among the substitutes.
Although the change came at late notice it should have handed Everton and Allardyce an avenue to attack.
Scarcely entrusted to start in his natural position at centre-half, Klavan at left-back with minimal preparation was there to be got at – especially with the pacy Theo Walcott and attack-minded Séamus Coleman directly up against him.
Yet Walcott was a peripheral figure throughout and spent more time trying to come inside when he would have been far more effective running at Klavan. In fact, as the touch map (below) shows, the Estonian (orange dots) had far more joy going forward.
The one time the former Arsenal man finally ran at Klavan he clipped a delightful cross to the back post which Tosun nodded agonisingly wide. Why it took him so long to do so is anyone's guess.
Rooney STILL not a midfielder
Hauled off after 57 minutes, you did not have to be a body language expert or lip reader to work out what Wayne Rooney felt of Allardyce's decision to hook him once more.
The former England captain was reluctant to shake hands with his boss and could clearly be seen to mouth the word ‘bulls***' as he took his place on the Everton bench.
But Allardyce was right to withdraw him. Rooney was slow, ponderous and unable to affect the game positively. He lost possession twice, completed just 71 per cent of his passes and won only two tackles. Rooney can only get away with playing in midfield against inferior opposition and Liverpool, even with five changes, aren't that.
But while he is earning top dollar and under contract at Goodison Park for another year the Toffees manager must find a use for him. Just not in the centre of midfield.
Big Sam loses bottle
For a man who has made a career out of bloodying the noses of bigger teams, Allardyce's approach to this game was as surprising as it was disappointing.
A week ago the former England boss sent his Everton side out to go toe-to-toe with champions-elect Manchester City. Yet today, against an unfamiliar Liverpool side whose priorities lay elsewhere, Allardyce was content to simply avoid defeat.
Already a divisive figure at Goodison Park thanks to his direct style and downplaying of expectations, this was Allardyce's chance to pick up a showpiece result – Everton haven't beaten Liverpool since 2010 – and prove he is the man to lead the Toffees next season.
On the face of it a front three of Walcott, Tosun and Bolasie suggested the Toffees would look to unsettle their rivals. Instead Everton set up with a low block, got men behind the ball tried to ensure they weren't beaten.
It wasn't until the final 15 minutes that the handbrake came off and in that period the Toffees had three gilt-edged chances to win it.
Regardless of whether Allardyce leaves this summer or next season his reign will be remembered with little fondness.