Saints, meanwhile, stay five points clear of the drop zone after their two-game winning run was ended.
City have a game in hand but will go into their next league game, at Crystal Palace next weekend, playing catch-up.
"It was never in doubt that he was a great player in my side", said the German at his post-match press conference.
Salah has been criticised in recent weeks for being too selfish in front of goal in desperation to end his scoreless streak.
Starting XI: Alisson (6); Alexander-Arnold (5), Matip (5), Van Dijk (6), Robertson (5); Wijnaldum (6), Fabinho (6), Keita (7); Salah (7), Firmino (6), Mane (7). "But he takes the responsibility", Carragher told Sky Sports.
Liverpool have already collected three more points away from home than they matched after last year's full season, where they lost and drew five games and conceded 28 goals. The performance was not at the top level but at this stage it is about fighting. "People want us to play like Manchester City". That is massive in this insane league and everyone is waiting for us. "But I'm still very proud".
Oleksandr Zinchenko reassures Manchester City over 'not so serious' injury
One of those, Sergio Aguero , will be assessed on Saturday morning to determine whether he is fit enough. Now it's the business end and I just need to do what I can to help the team win games.
Salah's record-breaking debut campaign was capped by the Premier League player of the year award - only Suarez and former England global Michael Owen had previously won it as Liverpool players. "We are in the race and that is nice".
However, the equaliser arrived before half-time with Naby Keita picking the flawless moment to score his first goal for the Reds, heading past the reach of Angus Gunn. After that we controlled it more. We've got to win every game.
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As for Southampton, they find themselves at No. 16 on the table, but still run the risk of being relegated should they suffer a late season collapse.