Harding's goal came after 25 minutes when Rickie Lambert and Jason Puncheon combined to leave the former Leeds player with a tap-in from six yards for his third of the season.
Southampton, who still hold hopes of reaching the play-offs, had plenty of chances to add to Harding's third goal of the season.
Radhi Jaidi was unmarked when he headed over and Puncheon hit the outside of the post after 20 minutes. Andy Hughes cleared off the line from Lee Barnard as Leeds struggled to stay in touch.
Whites manager Simon Grayson attempted to change things at half-time, sending on Leigh Bromby to bolster their defence.
Leeds were more competitive after the break and the nearest they came to snatching an equaliser was a stoppage-time shot from Robert Snodgrass which went narrowly wide.