Coach Jurgen Klopp's team, which started the day equal on points with Aston Villa, has 45 points from 20 matches.
Aston Villa has 42 points, while Manchester City, who played 19 matches, ranks third with 40 points.
Newcastle ranks ninth with 29 points.
The 31-year-old Salah, who scored 151 goals in the English Premier League with Liverpool, changed his shoes at halftime, and this had a magical effect.
With his two goals today, he tied with City player Erling Haaland as the top scorer in the competition with 14 goals. Salah also shares the top spot on the assists list with Aston Villa player Ollie Watkins, with eight assists each.
Liverpool controlled the match and Salah, who missed a penalty kick in the first half, finally broke the deadlock in the 49th minute when Darwin Nunez passed the ball to the Egyptian international and he finished it into the net from close range.
Alexander Isak scored against the run of play with only Newcastle's second shot in the match in the 54th minute when he received a through pass from Anthony Gordon and put it past goalkeeper Alisson.
But Curtis Jones put the hosts back in front in the 74th minute after a cross from Diogo Jota and Cody Jakpo scored four minutes later.
Sven Botman reduced the deficit for Newcastle in the 81st minute, before Salah, who was playing his last match before traveling to participate in the African Cup of Nations, scored from a penalty kick after Jota was fouled to seal the victory in the 86th minute.
Liverpool were the better side against Newcastle from the start, and goalkeeper Martin Dubravka kept the visiting team in the match with his wonderful saves.
Liverpool made 18 attempts in the first half, compared to one for Newcastle, which is its largest number in the first half in a Premier League match since April 2016 against Everton.
“It is a great result for us,” Salah told Sky Sports. “The match was very difficult and we were able to get three points and now we are at the top of the standings. We need to be calm and win every match.”
“The players spoke in the locker room (at halftime) saying we have to be calm. I did not want to travel to play with the national team with this performance (in the first half).”
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