On-loan Newcastle wing back Jetro Willems unexpectedly put the visitors in the lead with a sublime strike seven minutes in, but a first-half double from Mane followed by Salah's solitary goal in the second half made sure the Reds took all three points.
Here are some key talking points from an eventful day on Merseyside:
Firmino continues to dazzle
The goals were scored by the two usual suspects, but all eyes were on the remarkable Roberto Firmino as he showcased how integral he is to Liverpool's gameplan.
The South American star didn't feature in the starting eleven, benched for Divock Origi for rotational purposes. However it wasn't long after Mane had equalised for the home side that he was called into action, with Origi picking up an ankle injury just over half an hour into the game.
Instantaneously the number nine alerted the Magpies' back line, offering a dynamic that, arguably, no other striker in the world can provide. You only have to look at the Reds' second goal to witness the determination of Firmino, delving deep into the middle of the park to dispossess the helpless Christian Atsu, feeding the ball through to strike partner Mane who made it past Martin Dubravka to make it 2-1.
The next half brought an even better display from the Brazilian, the highlight of which coming in the 72nd minute. Salah fed the ball to the striker on the edge of the Newcastle box, who fractured the back line with a mesmerising piece of skill to return the ball to the Egyptian who calmly slotted the ball home in front of the travelling supporters.
It has almost become commonplace to the Liverpool faithful that this man will captivate the audience every week, and it's not just for show. The fan favourite is aggressive and hostile to opposing defences, but is yet so benevolent and charitable to the players around him in the final third, a rare blend for a forward in the modern game.
If there is anyone out there who is perplexed at the role that he offers to Jurgen Klopp's side, today's performance should be an indicator of his significance and right to be Liverpool's number nine.
Loveable Mane's confidence is sky high
It has been a delight to witness Mane's transition from the unknown debutant at Southampton to the player he is today, and it was expected that the Senegalese international would progress even further this season after picking up a Premier League golden boot last campaign.
The 27-year old grabbed two goals today, the first his most impressive.
Liverpool had struggled for the first 20 minutes or so to get a foothold of the game, Newcastle had scored with their first attempt on goal and like so many other sides at Anfield, planned to sit deep and invite pressure with hopes of travelling home with a point or three.
Unfortunately for the North East club that didn't go to plan, with Andy Robertson brushing past Emil Krafth to play the ball into Mane on the edge of the box, who stopped the pace of the ball and curled his effort into the top right corner.
He is instilled with passion too. As soon as the ball had hit the net it was not about celebrating, it was about resuming the game as quick as possible to hit the net again - instantly running back to the Liverpool half whilst signalling for the crowd to generate more atmosphere.
The second isn't as aesthetic as the first, but Mane's intelligent running made it easy for Firmino to feed the ball through to put him one on one with Dubravka. Although the goal was ultimately provided through the Slovakian goalkeeper's error, it put his side in the lead the and settled the nerves on the Kop.
No player in the Premier League has scored more goals than him in 2019 (18), and he is now the only Premier League player in history to play the first 50 home games for a single club without defeat.
Another fan favourite like Firmino, Mane's ruthlessness and unmerciful finishing is quickly making him one of the best players in the world.
Origi better suited to a central role
It's harsh to discredit the Belgian's performance today after playing only 37 minutes of the game, with the majority of them coming after he had picked up an ankle injury before being taken off, but the cult hero is quite evidently more comfortably suited to his natural striker role.
Before Firmino's introduction Liverpool's front three were constantly rotating between each other, perhaps to confuse a deep lying Newcastle defence. Both Salah and Mane were taking turns to move into the main striker role, whilst Origi was predominantly on the left wing.
Reds fans wont be surprised by this, having witnessed the former Lille and Wolfsburg man play wide sporadically throughout his career at Anfield, but he hasn't particularly captivated many whilst there.
Origi is a tremendously hard-working player whenever he puts on a red shirt, and he does posses the power and pace to cause a rift in back lines. But one thing that he lacks that Salah, Mane, and even Xherdan Shaqiri have, is the vision and creativity to produce on the wing. He struggled to create any chances for himself or anyone else today, and doesn't seem compatible with the overlapping of Robertson on the left hand side.
The Champions League winner is best playing on the shoulder of centre backs, using his height to his advantage (he beat Jamaal Lascelles in the air to score a last minute winner in the same fixture four months ago) and using his energy and vibrancy late in games to scare tiring defenders into a mistake. It is not a criticism of the 24-year-old, merely a perception of his skillset and where he is best suited.
Buoyant morale for Reds
After match week five of the Premier League, the red half of Merseyside can feel upbeat and excited at what's awaiting them in the upcoming weeks.
Alex Oxlade-Chamberlain started at Anfield for the first time since April 2018 after a long recovery from a ruptured ACL, and although it will take a few weeks to begin to hit top form, there were a few promising signs for the England international. Today's game also the 26-year-old's 50th appearance for the club.
This weekend's performance boosts the already soaring Reds confidence ahead of Tuesday, when Liverpool travel to Naples to face Napoli in their opening fixture in their Champions League defence. Win that, and Klopp and his men will be in high spirits for Sunday's clash against Chelsea.
Despite Alisson's injury putting him out of action for a lengthy spell, Naby Keita is set to return from injury within days, and the Carabao Cup match against MK Dons could see him and many other fringe players get some well-needed game time.
Along with all of this, Norwich City's emphatic 3-2 victory against title rivals Manchester City at Carrow Road later on in the day has given Liverpool a five point gap at the top of the table, setting the pace early on for what looks to be another exhilarating campaign.