Liverpool finally beat a side managed by Rafa Benitez, and Jurgen Klopp got his first victory over Newcastle United at the fourth attempt.
The reds were simply too strong against a defensively disciplined Newcastle side at Anfield.
Salah and Mane grabbed the goals - but a crucial save by Karius prevented an equaliser on the stroke of half time that could have affected the direction of the game.
There were two changes in the Liverpool side following last weekend's 4-1 win over West Ham.
Joel Matip and James Milner made way - with Dejan Lovren and Jordan Henderson returning to the Liverpool line up.
Newcastle were without Jonjo Shelvey who missed the game with a knee injury.
Before the match both Benitez and Klopp had identified that Newcastle's strategy would be a defensive one - aiming for counter attacking opportunities.
And Newcastle certainly stuck to the script for most of the first half - they were well organised with a back 5 protected by a middle 4 with Dwight Gayle providing an outlet up front.
They frustrated Liverpool's determination to move the ball forward. Mo Salah hesitated too long on a rare first half chance allowing defender Lejeune to get back and intercept.
A Liverpool corner was completely missed by the Newcastle keeper but there was no Liverpool player close enough to react, and Sadio Mane and Dejan Lovren also had opportunities to give the reds a first half lead.
But with five minutes remaining in the first half, Mo Salah broke the deadlock with his 24th Premier League goal - his 32nd in all competitions.
From a midfield mistake, Alex Oxlade-Chamberlain powered forward, half-feinted too shoot before feeding the ball across to Salah. This time the Egyptian didn't dwell too long - stopping the ball with his right before firing between the keeper's legs with his left.
There was a late scare however as first half injury time ticked down. Diame struck a sweet shot from just outside the area - heading to the top right hand corner - but Karius reacted quickly and produced an excellent one handed save to keep Liverpool in the lead.
The second half was slightly more open. Although Newcastle's main priority remained defending their midfielders were slightly more advanced as they had to keep an eye on trying to get back into the game.
But 10 minutes into the second half Liverpool doubled the lead. In a more open second half - it was Sadio Mane who fired in after a neat threaded ball by creator in chief Roberto Firmino.
The game was effectively over and shortly after the substitutions for both sides began which killed off a lot of the poise in the game.
Newcastle have struggled for goals and at 2-0 even they appeared to know there was no way back.
There was late incident with Mo Salah involved in two potential penalty appeals. The second one showed an offence was committed outside the area but the Newcastle defender got away with the decision which would have caused him to receive a red card for denying a goalscoring opportunity.
Fortunately for the officials, the result wasn't altered by the poor decision.
The reds can now look forward to the Champions League clash with Porto! A 5-0 first leg lead remember!
MOTM: Alex Oxlade-Chamberlain