The home side dominated from the off with wingers Anthony McNamee and Jon-Paul McGovern tormenting Saints full-backs Lloyd James and Dan Harding at will.
The hosts grabbed what turned out to be the game’s winner in the ninth minute after some trademark McNamee trickery down the right had earned a corner.
McGovern’s inch-perfect delivery found centre-back Sean Morrison who thumped a header past a sprawling Kelvin Davis in the Southampton goal.
The Robins should have been 2-0 ahead five minutes before the break when man-of-the-match McNamee’s whipped centre found Billy Paynter unmarked five yards out, but his diving header was superbly saved by Davis.
The Swindon onslaught continued throughout the second period and despite failing to get the second goal their fine play deserved, an unlikely point for the visitors never looked like coming.