Botti Biabi (pictured above right) scored twice, once either side of half-time, before Kenji Gorre and Adnan Maric (pictured above left) put them 4-0 up.
Bristol City rallied late on, with Ben Cabango scoring an own goal before substitute Cameron Allen netted the Robins' second.
But Swansea held on to set up a home tie with Brighton & Hove Albion in the quarter-finals.
Swansea: 13 Apr v Arsenal (A) PL2
Bristol C: 13 Apr v Sheff Wed (A) PDL