Lambert only arrived at the St Mary's Stadium on Monday following his transfer from Bristol Rovers where he scored 29 league goals last season.
And he signed no signs of slowing down with a predatory finish on 29 minutes before midfielder Adam Lallana grabbed the second on 68 minutes.
Lambert already fired a warning shot to the Cobblers when he fired a 25-yard freekick just wide.
But the first real sight of goal fell to visiting midfielder Ben Marshall.
The Saints failed to clear their lines and Marshall burst into the area only to find Kelvin Davis – who was one of the highly-rated players to pledge their loyalty during Southampton's troubled period – out of his goal in a flash to smother his effort.
Marshall again tested Davis with a freekick that bounced just in front of the keeper but he clung on as the Cobblers upped the ante.
Frontman Steve Guinan, who replaced Adebayo Akinfenwa in the starting XI, fired a volley just over and Andy Holt should have worked the keeper with a header from a corner.
And they were made to pay for their prolificacy when the Saints broke the deadlock just before the half-hour mark.
Lloyd James found himself in space on the right and whipped in a cross that Lambert finished off.
Guinan missed a golden opportunity to draw the visitors level when he just failed to get the right connection on a ball into the box.
Southampton boss Alan Pardew opted to throw Polish striker Gregor Rasiak into the fray and the decision immediately paid dividends as his knockdown allowed Lallana to tap in from close range and book the home side's place in the next round.