The Egypt international had failed to score in his last eight games in all competitions ahead of their trip to the south coast.
Salah turned provider in the first half when the 2018 PFA player of the year teed up Naby Keita for his inaugural Premier League goal.
With 10 minutes to go, Liverpool FC appeared to be on the brink of dropping two vital points in the Premier League title race.
However, Salah launched a counter attack from inside his own half before the former Chelsea FC winger held his nerve to place his finish past Angus Gunn despite a lengthy run.
Substitute Jordan Henderson extended the away side’s lead with another goal on the counter attack with four minutes left to play.
Former Liverpool FC defender Carragher could barely contain his excitement at Salah’s crucial goal.
“As he is running wth the ball people are saying, ‘pass, pass’ especially with the form he has been in. But he takes the responsibility,” Carragher told Sky Sports.
“He’ll be the man who puts Liverpool back at the top of the league.
“It goes from one end to another. The charge is on. It’s a great finish.
“As he breaks, has he got the pace to go past the defender? Has he got the confidence? Will he play Firmino in? No [he does it himself]. How big could that goal be? There have been so many late goals.”
Salah has now scored 18 goals in 32 games in the Premier League this season after ending an eight-game drought at St Mary’s.
Liverpool FC moved two points ahead of defending champions Manchester City at the top of the Premier League table thanks to their 3-1 win at Southampton.
The Reds will return to Premier League action with a home clash against Chelsea FC at Anfield on Sunday 14 April.
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