The breakthrough came after 40 minutes when Alex Oxlade-Chamberlain drove forwards and slipped the ball to Salah, who took a touch to control before firing between goalkeeper Martin Dubravka's legs.
Stressing that his new manager has played a pivotal role in helping him to improve his overall performances, Oxlade-Chamberlain will delight Liverpool fans with his belief that there is still "more to come".
In particular, the first Liverpool goal scorer, Mohammed Salah, was something of a formality as the Egyptian scored for the seventh consecutive game.
His hard work got Liverpool the ball back to break and score, and his flicked pass to Mane was exquisite.
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The loss leaves the Toon Army just two points clear of the relegation zone, but the determination and discipline that they showed in this match will hold them in good stead for the fight to stay in the top flight. "He's such a good runner off the ball, he makes such good runs, and it makes it easy for me or whoever to find him".
"If you've got Championship players you probably need to tell them two or three times to do something".
Newcastle's determined defending was denying Liverpool space, but Salah dallied when he could have fired at goal, Lejeune recovering to make a block. "We feel comfortable at Wembley, we know how we play here".
With nine league games and, in all likelihood, at least three European ties left in Liverpool's season, Salah should have the 36 goals scored by Robbie Fowler, 22 years ago, well within his sights.
A ricochet in midfield fell to Oxlade-Chamberlain, who fed Salah for the Egypt worldwide to take a sharp touch out of his feet and hit a crisp finish through the legs of Dubravka.
Mane's second half strike ensured there was no joy for Newcastle manager Rafa Benitez. With AP PhotosLiverpool and Tottenham are putting the squeeze on Chelsea in the race for Champions League qualification next season after chalking up 2-0 victories in the English Premier League.
MF Emre Can, 6 - He got through most of the match on autopilot and played one long pass to Mohamed Salah that would ordinarily have ended with a goal. What do you think?
Newcastle United lost to Liverpool at Anfield in the Premier League on Saturday.
For the second week running Liverpool needed patience in the first half against a deep-lying side. That's exactly the life of a top goalkeeper at a top team.
"The best team we had at this time was the team that finished second in the league with 86 points".