Striker Lambert, signed by Saints boss Alan Pardew for £1million from Bristol Rovers on Monday, met Lloyd James' fine cross to give the south coast side a 29th-minute lead.
Highly-rated midfielder Adam Lallana nearly doubled the home side's advantage minutes later when his well-struck half-volley was kept out by Northampton goalkeeper Chris Dunn.
Cobblers striker Steve Guinan missed a fine chance before half-time and was made to pay for his poor finishing as the tie was killed off after 68 minutes.
James' free-kick was headed back across goal by substitute Grzegorz Rasiak and Lallana was on hand to tap in from close range.
Northampton were forced to chase the game but could not break through a resolute Saints defence - and the home side went close to adding further goals first through Rasiak and then fellow substitute Jake Thomson, whose late drive was well saved by Dunn.