win since succeeding Nigel Adkins at St Mary's and left City facing the prospect of falling 12 points behind Manchester United in the race to retain their crown.
And Mancini admitted afterwards they will have no chance of catching Alex Ferguson's side if they can beat Everton at Old Trafford on Sunday.
"Maybe it's over. I think we now have a 10 per cent (chance of winning the title). I don't think United will drop 12 points," he said.
"(The performance) was worse than poor. We didn't play. We only played with two players in this game and 11 against two is difficult to win.
"It's very strange because we played very well for the whole of January so I don't know why we did so badly today. Two of the goals we conceded were incredible for a top team."
Having already been dumped out of the Champions League in the group stages, pressure is now mounting on Mancini.
The absence of Carlos Tevez from the matchday squad due to "family reasons" meant that Edin Dzeko and Sergio Aguero were the only two strikers available but even the return of Yaya Toure from Africa Cup of Nations duty couldn't inspire a performance.
The writing was on the wall for City inside the first seven minutes as Jay Rodriguez's effort was saved by Hart, only for Jason Puncheon to slot home the rebound.
The day got even worse for Hart midway through the half when he failed to hold a shot from Rickie Lambert and Steven Davis pounced to make it 2-0.
"It was a really bad mistake," admitted Mancini. "It's not strange because he did two or three mistakes this year and this is not good for us. Joe is a good keeper but a keeper cannot make mistakes like this. It cost us the second goal."
Dzeko's goal six minutes before half-time at least got City back into the game but they were lucky to escape further punishment when Toure brought down Jay Rodriguez in the box, only for referee Martin Atkinson to wave away their appeals.
But any hopes of a comeback were ended three minutes into the second half when Barry turned a harmless cross from Lambert past Hart.
It was a disastrous moment for the England midfielder, although City's reaction to falling two goals behind again will also have worried Mancini.
A lack of attacking options on the bench saw them switch formations several times as they searched for a way back into the game but the Italian manager acknowledged that many of his players are living off past glories.
He said: "We need to forget what we did last year. If you want to stay at the top for a long time you have to work hard."
For Pocchetino, who succeeded Adkins last month, the result was very welcome as it lifted Southampton six points clear of the relegation zone.
And the former Espanyol manager admitted it had been an "incredible day".
"I am very happy for the team and the players who deserved the victory," he said.
"It's been an incredible day. There are things that we can improve on and with time we will get better. Three points are very important but the added value is that we have shown we can play well and win matches against great teams."