A second-half error from Swindon goalkeeper Phil Smith gifted Leeds three points, but United's win deserved huge credit after the club were forced to complete 82 minutes with only 10 players.
United were reduced to 10 men in the eighth minute when Sheehan was sent off for a two-footed tackle on Jon-Paul McGovern, but the visitors silenced the crowd at the County Ground by opening the scoring 15 minutes later.
Jerel Ifil fail to cut out a low cross from Luciano Becchio, leaving Jermaine Beckford unmarked to drive a clinical finish into the top corner of the net and claim his ninth goal of the season.
United came through the first-half with unexpected ease until the 45th minute when Anthony McNamee's curling cross found Simon Cox, whose volley from six yards flashed beyond David Lucas.
But Leeds moved back in front when Swindon goalkeeper Phil Smith fumbled Neil Kilkenny's free-kick into his own net, and a sensational save from Lucas – keeping out a close-range header from Cox – helped Leeds hang on for a priceless win which was sealed by Beckford five minutes from time.