After taking a fifth minute lead through debutant Josh Morris' first goal for the club, City could have added to their early lead as they dominated the first period at the County Ground.
Billy Clarke saw a penalty saved and an ambitious lob from distance narrowly go wide, while Billy Knott and Christopher Routis also went close in the opening half of the season.
Swindon roared back into the contest after the break, however, with a Nathan Byrne hat-trick and a further strike from Jonathan Obika eventually making it a difficult afternoon for the Bantams.
City supporters can watch Phil Parkinson's full post-match Swindon interview via Bantams PlayerHD - please click HERE.