Liverpool were well worth their lead but Southampton were a different proposition after the break. Mane had a penalty saved by Simon Mignolet in the 49th minute but he made amends with a close-range finish just past the hour mark.
Christian Benteke should have restored Liverpool's two-goal advantage but shot wide from a good position.
From then on it was all Southampton as the visitors, looking heavy-legged after their midweek Europa League clash with Manchester United, sat back in the hope of playing out time.
Graziano Pelle shook off the attentions of both Liverpool's centre backs to equalise with a powerful 20-metre strike seven minutes from time.
Mane snatched the points three minutes later with a precise low finish after Mignolet had failed to clear and the ball was played back into the danger area.
Victory moved Southampton into seventh place on 47 points, three more than Liverpool.
"I do not know about the top four, we are not closer than we were before. There are nine games left, that is 27 points, it is possible, I don't know," Liverpool manager Juergen Klopp told the BBC.
(Reporting by Martyn Herman; editing by Clare Fallon/Toby Davis)
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