Liverpool needed a Hugo Lloris error to beat Tottenham 2-1 last time out and they rode their luck again at St Mary's Stadium on Friday, Jurgen Klopp's side showing their nerves.
Shane Long's 50th Premier League goal had put Southampton into an early lead but Naby Keita's first strike for Liverpool ensured the scores were level at the interval.
Salah's dry spell has been well-documented but he ended his eight-game wait for a goal in the 80th minute, embarking on a long solo run before finishing superbly.
And captain Jordan Henderson then made sure of the win for the Reds with four minutes to go by turning home a Roberto Firmino low cross, as Klopp's men went two points clear at the top, albeit having played a game more than City.
Liverpool made a slow start and Southampton led in the ninth minute.
Ryan Bertrand was given too much time and space to deliver a left-wing cross that Pierre-Emile Hojbjerg flicked on at the near post, with Long left unmarked to sweep home.
Sadio Mane directed a free header straight at Angus Gunn in response before Virgil van Dijk turned another dangerous Bertrand cross over his own crossbar.
Gunn was beaten in the 36th minute, though, Liverpool keeping Southampton pegged in and Keita heading home Trent Alexander-Arnold's cross at the end of a sustained period of pressure.
James Milner and Henderson had been left out of the starting line-up by Klopp but he turned to the duo early in the second half as Liverpool struggled to break Southampton down.
Keita appealed for a penalty but appeared to go down easily under Maya Yoshida's challenge, with Liverpool needing Salah to break his drought to nudge ahead.
The goal came from a Southampton corner, with Henderson sending Salah on a long run through the Saints half, the Egypt forward keeping cool to beat Gunn from the edge of the box.
Southampton, not yet sure of avoiding relegation, had to throw men forward in search of an equaliser and Liverpool duly killed the game with a clinical break, Firmino teeing up Henderson as the Reds moved back above City.
Pressure back on City
As Liverpool struggled in the second half, it seemed like the advantage would go to City.
But instead, Pep Guardiola's side will have to wait until a week on Sunday to respond to Liverpool's latest late show, when the defending champions go to Crystal Palace.
The title race looks set to go to the wire and the fight against relegation could do the same, with Southampton still only five points clear of the bottom three after surrendering their lead.
