Published: 2019-08-17 23:06:40 BdST
Liverpool have six points from two games after their opening 4-1 home win over Norwich City. Either champions Manchester City or Tottenham Hotspur, who clash in the 1630 GMT kick off, can leapfrog them.
The European champions looked weary in the opening 30 minutes after their penalty shootout win over Chelsea in the UEFA Super Cup in Istanbul on Wednesday but Mane fired them ahead against the run of play.
The lively Senegal forward teed himself up on the edge of the penalty area on the stroke of halftime and curled a rasping shot past home goalkeeper Angus Gunn, who was left clutching at thin air.
Liverpool dominated after the break and missed several chances before Firmino picked up a loose ball, side-stepped a defender and drove a crisp low shot past Gunn in the 71st minute.
But the visitors barely hung on in the closing stages after a howler by their stand-in goalkeeper Adrian gifted the home side a lifeline.
The Spaniard's poor clearance rebounded off Danny Ings into the back of the net and the former Liverpool striker missed a gilt-edged chance to snatch a late draw for the Saints when he skewed wide a close-range shot in the 87th minute.
Mane acknowledged the exertions of playing last season's Europa League winners Chelsea in the Turkish capital took its toll on Liverpool.
"After playing 120 minutes on Wednesday it was not easy," Mane told the BBC.
"But we got the three points and I think we deserved it. Once we scored the first goal it was easier for us."
Mane had mixed emotions about scoring against his former club and conceded Liverpool were rattled by Adrian's blunder.
"I'm sorry I have to score but I'm wearing a Liverpool shirt now. Southampton was a very good step for me, I have big respect for this club.
"(After Southampton's goal) I was nervous because it wasn't over. We had another 10 minutes to go. It can happen, it is part of football."