Southampton took the lead on 17 minutes with a close range header by Do Prado, following a cross from Lee Holmes.
Forte then made on 2-0 on 30 minutes after Phil Smith failed to save his shot.
Swindon went on to look a lot better in the second half, and pulled a goal back on 84 minutes with a free kick from the edge of the area scored by Mehdi Kerrouche to make it 1-2.
Swindon had chances to get a late equaliser with both Flint and Clarke missing chances.
However it was Southampton who scored the late goal on 94 minutes when Lambert took the ball around Smith to score into the empty net making it 3-1.