Both teams had quality chances in the opening 30 minutes, but Liverpool would take the lead when Mane headed home a pinpoint Mohamed Salah cross after Alisson had made a great save on Southampton's Che Adams seconds earlier.
Jurgen Klopp's Liverpool took the 1-0 lead into the break with some splendid goalkeeping from Alisson helping keep Southampton off the scoresheet in the first half.
Southampton pushed for an equaliser, but Alisson continued to stand tall in the Liverpool goal and the visitors were unable to level the score.
Thiago added a classy second for the home side just before full-time, scoring from the top of the penalty area to reach the final scoreline.
"It is an amazing feeling [to score] but the most important thing is the three points. We keep fighting for a small miracle," Thiago said of Liverpool's top-four hopes.
"The window is open so we try to go through the window."
Manchester City's earlier defeat by Chelsea meant that, technically, Liverpool are still champions. But they continue to look nothing like the marauding machine that took them to the title last season.
Southampton keeper Fraser Forster was kept busy with saves from Diogo Jota and Salah but Liverpool looked uneasy whenever Southampton attacked and were lucky not to fall behind.
Nathan Tella found himself in acres of space down the right flank and surged forward before crossing low for Adams who looked certain to score before Alisson smothered his shot.
Liverpool made the most of that let-off a minute later when Salah hoisted the ball into the box and Mane headed in.
Tella wasted a chance to equalise before halftime when he headed straight at Alisson, while Adams had the ball in the net only for his effort to be disallowed for a foul.
Jota forced a sharp save by Forster after half-time while at the other end Alisson was fortunate when he cleared the ball straight to Adams, whose shot allowed the keeper to recover.
Mane had a goal disallowed but Thiago settled the issue when he threaded a shot into the corner.
The result keeps Liverpool's Champions League hopes alive, although there's work to do as they sit on 57 points, six behind fourth-placed Leicester City but with a game in hand.
Southampton remain in 16th place and, while they are not mathematically safe from relegation, they look like hanging onto their top-flight status.
Information from Reuters was used in this story.