The reigning champions then became their own worst enemy as Steve Davis doubled the home side's lead after City goalkeeper Joe Hart allowed Lambert's shot to squirm through his hands.
Edin Dzeko reduced the arrears before half-time but a Gareth Barry own goal early in the second half secured a first win for Saints' boss Mauricio Pochettino and left Mancini ruing his side's performance.
Asked if it was a poor performance Mancini replied: "No, I don't agree, it was worse than poor.
"We didn't play. We played with maybe two players in this game and with 11 against two it is difficult to win.
"Probably we think that, because we won last year, we are top players but if you want to stay on the top for a long time you have to work always and forget what you did last year."