Lambert contributed a fine display to help inspire the victory, which was sealed by first-half strikes from Jason Puncheon and Steven Davis and an inexplicable Gareth Barry own-goal after the break, with Edin Dzeko scoring City's only goal.
For Southampton the result marked a first success under new manager Mauricio Pochettino and Lambert believes they were worthy winners.
"I think it's our best performance (of the season)," he said on ESPN. "We were absolutely superb today and got exactly what we deserved.
"We looked for the early goal and looked to put them under pressure early and hope to capitalise on it. It worked perfectly. Then we went 2-0 up, which was even better."
The margin of victory could have been even greater had referee Martin Atkinson not waved away Jay Rodriguez's penalty appeals on the stroke of half-time.
Man of the match Rodriguez said: "I ran in and I think there was a little nudge, but the referee hasn't given it so we got on with the game."
He added: "We press hard, take the game to them. As you can see, it creates chances and puts us on the front foot. We've taken to it well and put in a great performance."
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