A first-half short on quality and final product saw Everton start fast to the cacophonous chorus of the home crowd, before Liverpool grew into the game - with neither having a shot on target before half-time.
The encounter picked up after the break, Roberto Firmino squandering a good one-on-one opportunity as he tried to lift his effort beyond Maarten Stekelenburg.
Everton should have been down to 10 men on 68 minutes when Ross Barkley went through Jordan Henderson with a nasty challenge, though he was awarded only a yellow.
Liverpool enjoyed the better of the second-half but struggled for chances across an underwhelming 227th Merseyside Derby, Firmino seeing a volley well saved by substitute goalkeeper Joel Robles.
Yet in the final moments of stoppage-time, Daniel Sturridge's strike came back off the post and out for Mane, who reacted faster than any Everton defender to tap into an empty net and seal Liverpool's first 1-0 league win in the calendar year of 2016 and their first win by that scoreline at Goodison sine April 1978.
Their efforts at the opposite end of the pitch also mean Liverpool have kept four clean sheets in their last six league games after just one in their previous 11.
Everton remain ninth in the table as their six-year wait for a victory over Liverpool continues.
Reds unchanged as Blues swap Jagielka for Funes Mori
There was only one change between the two teams when the teamsheets were revealed an hour before kick-off.
That change came from Ronald Koeman, who was forced to bring in Ramiro Funes Mori - sent off in the last meeting between these sides - for Everton captain Phil Jagielka after his red card in Tuesday's win over Arsenal.
Otherwise, they kept with a familiar line-up - Aaron Lennon and Enner Valencia flanking Ross Barkley behind lone striker Romelu Lukaku.
Jürgen Klopp stuck with an unchanged eleven for Liverpool, with Divock Origi once again up-front on the back of scoring five goals in as many games.
Ragnar Klavan and Simon Mignolet kept their places having come in for the mid-week win over Middlesbrough, while Sturridge and Emre Can returned to the bench with Joel Matip still sidelined.
Everton fail to take advantage of superior early spell
The hosts' game plan was evident from the off, looking to play off the physical presence of Lukaku up top.
Everton started off the better side, the white-hot atmosphere inspiring an impassioned first 15 minutes in which 72.5 per-cent of the game was spent in Liverpool's half.
Yet Mignolet remained untroubled in goal, Barkley firing a low 20-yard drive wide a few minutes after Klavan's outstretched right foot prevented Lukaku from turning in a Leighton Baines cross.
Having weathered the storm without conceding, Liverpool began to come into the game around the half-hour mark as they started to find the composure and control they so desperately lacked earlier on.
That paved way to some much improved play in possession, and their first chance came through a well-worked team move that started on the left and moved to the right, Origi firing wide first-time from Nathaniel Clyne's ball across the box.
Everton's first corner-kick of the evening should have resulted in more, their two goals in mid-week having come from set-pieces, but Funes Mori headed wide from Barkley's corner.
The lack of chances - and the fact neither side could manage a shot on target - encapsulated a first-half short on quality despite plenty of industry and endeavour.
Firmino wastes game's best opportunity yet
Swansea's clash with Sunderland was the only other Premier League this season to be without a shot on target by the half-way mark, the Swans eventually winning 3-0. And the feeling here was that things could only improve.
That elusive first shot on target came through Liverpool after 49 minutes, when Roberto Firmino should arguably have handed the Reds the lead.
The Brazilian spotted a gap in Everton's back-line and ran on to Milner's ball over the top, dismissing the attentions of Funes Mori only for Stekelenburg to save well as he looked to lift his shot over the 'keeper one-on-one.
The looping deflection of Firmino's shot nearly fell for Mane and Lallana, neither of whom could get a clean connection, before Clyne's well-struck effort was blocked.
Lukaku's backwards header from half-time substitute Gareth Barry's ball had Mignolet briefly worried, the burly striker forcing Lovren into the book soon as he threatened to sprint clear of the Croatian.
Firmino was at the heart of another promising Liverpool move on 57 minutes, flicking a through ball to Mane with a clever backheel, although the Senegalese winger's pace couldn't quite carry him to meet the loose ball before Stekelenburg.
The follow-through of that challenge left Stekelenburg in serious discomfort, with Ashley Williams temporarily forced to overtake goal-kick duties before Joel Robles took the limping Dutchman's place.
That stoppage gave Everton time to collect having been inferior since half-time, though perhaps the introduction of teenage striker Dominic Calvert-Lewin - who was ready to come on before Stekelenburg's enforced withdrawal - would have been of greater benefit.
Barkley lucky not to see red for nasty challenge
Barkley was soon fortunate not to be more harshly punished for his horribly late challenge on Henderson as the game ticked on, the midfielder catching his England teammate high on the ankle with a studs-up sliding tackle that prompted a melee between both sets of players.
With Liverpool increasingly dominant, Koeman finally opted to hand 19-year-old striker Calvert-Lewin his second league appearance with less than 20 minutes to go.
Klopp, otherwise, decided against substitutions as he kept faith with his starting side. That faith almost paid off after another good passage of play in the 75th minute, Lallana making his way past Williams to find Mane, whose driven cross resulted in Origi's side-footer being deflected wide.
Everton finally registered their first shot on target from a free-kick, Williams heading comfortably into the grasp of Mignolet, yet remained second-best.
Substitute goalkeeper Joel Robles was required to pull off a sublime reflex save with 10 minutes to go, Firmino meeting Milner's corner on the volley with a goalbound effort before Robles' intervention.
Sub Sturridge helps hand Mane the match-winner
Liverpool finally turned to the bench on 80 minutes, Sturridge and Can both coming on in place of Lallana and Origi, who failed to stretch his rich vein of goalscoring form to six in six.
Barkley was afforded further fortune by referee Mike Dean after kicking the ball away shortly after an Everton foul, before the manic ending to come, which included eight minutes added time.
In those final minutes, Liverpool finally struck the game's opening goal. Sturridge's left-footed shot from the edge of the box seemed tame, but it rebounded back off the post and into the path of Mane - who duly applied the finish to send the away end into sheer ecstasy.
It was soon almost two as Seamus Coleman had to lunge to deny Firmino on the line, but regardless - Liverpool stole the three points to ensure that they will spend Christmas in second - six points off of league leaders Chelsea.
Liverpool have only lost one of their previous 15 league matches, winning 10 and drawing four, inflicting Everton's first home defeat since March 19th, 11 games ago.