A first-half Tresor Kandol header was enough to give Leeds their first win under new manager Gary McAllister.
Kandol arrowed his effort straight into the top left-hand corner from Dave Prutton's cross in the 24th minute.
Simon Cox should have pulled Swindon level shortly before half time but headed over unmarked from three yards.
The visitors came close to doubling their lead but Swindon goalkeeper Phil Smith tipped over Prutton's corner and Jonathan Howson's five-yard volley.
Swindon: Phil Smith, Comminges, Aljofree, Kanyuka (Ifil 17), Vincent, Roberts, Easton, Peacock, Allen (McGovern 68), Paynter, Cox.
Subs Not Used: Brezovan, McNamee, Ashikodi.
Leeds: Ankergren, Richardson, Marques, Michalik, Kenton, Prutton, Howson, Kilkenny (Thompson 78), Johnson, Kandol, Beckford.
Subs Not Used: Lucas, Carole, Elding, Sheehan.
Goals: Kandol 25.
Ref: Chris Foy (Merseyside).