Goals by Nick Powell and Steven Fletcher kept Stoke's faint play-off hopes alive as they sentenced Bristol City to a seventh straight home defeat.
Powell netted his 12th goal of the campaign to give the Potters a first-half lead, finishing off Jacob Brown's cross.
Fletcher then sealed the points with a curling free-kick that zipped beyond Dan Bentley just after the hour mark.
Stoke's first away success in 12 games keeps them in 11th place, seven points off the top six, while the Robins remain 14th.
Bristol City handed 18-year-old Sam Bell his first league start - but the young forward's full debut lasted just a quarter of an hour before he was forced off with a knock.