Lambert opened the scoring in the 15th minute after strike-partner Papa Waigo wriggled clear of his marker before feeding a pinpoint pass to Lambert to slot home from close range.
Saints' lead was doubled on the half hour after a comical own goal from Dons defender Danny Woodards - an attempted clearance hitting him in the chest and dribbling past goalkeeper Willy Gueret.
Paul Ince's side pulled one back on the stroke of half-time through midfielder Mark Randall's neat 12-yard finish, but the home side remained on top.
The second half became a scrappy affair as the away side desperately tried to get back into the tie, but clear-cut chances remained at a premium even as Saints' defending became more nervous.
The impressive Adam Lallana eventually made the game secure and confirmed Southampton's return to Wembley - their first final there since 1992 - with a powerful finish in the 87th minute.