First half goals by Guly Do Prado and Jonathan Forte, together with a late strike from Ricky Lambert were enough to book Saints' passage into round three.
Swindon pulled a goal back late on through Mehdi Kerrouche but by then Southampton had given the home side a footballing lesson.
Swindon had the best of the opening 10 minutes but their final ball lacked quality and Southampton went ahead in the 17th minute when Do Prado took advantage of a good cross from Lee Holmes who had been given far too much room on the right.
Swindon's reply was an 18-yard effort from Paul Caddis which flew wide and then Leon Clarke did not make the best of an opportunity down the left.
Southampton went further ahead when Forte picked up a ball played through the left channel and although Phil Smith got a hand to the shot he could not prevent the ball from crossing the line.
Swindon won their first corner after 37 minutes and the half clearance fell to Lander Gabilondo. His low cross fizzed through the goal area with no one to get the last touch.
Swindon started the second half with more determination but could not prize open a tight visiting defence.
Swindon had more possession in the second half but Southampton continued to hold the upper hand and did enough to protect their lead.
Swindon pulled a goal back through Kerrouche in the final 10 minutes.
The Swindon substitute was fouled on the 18-yard line and he rifled the ball home to give the home side some hope.
But Southampton were not to be denied when they struck in the second minute of added time through Lambert who took advantage of a defensive slip to seal the victory and claim a third round spot against either Charlton or Preston.