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Bony came close himself shortly after when he met Jonjo Shelvey's cross into the box, but he could only head the ball narrowly wide of the target.
Southampton grew into the contest and should have taken the lead after 32 minutes when Nathaniel Clyne's whipped cross found Sam Gallagher, but the young striker failed to find the target with his header.
Pablo Hernandez picked out Bony as the game edged towards half time, only for the Ivorian to see his resulting effort soar over the bar.
In what was an uneventful second half, it took until the second minute of added time for any real opportunity, and it proved decisive in terms of a winner.
Ashley Williams failed to control the ball inside his own area as it flew high above the back-peddling Vorm and eventually came back off the crossbar, where Lambert was waiting to tap it over the line with his shoulder.
Southampton remain in eighth place following the victory in South Wales, while Swansea are 13th.