Hosts Crawley dominated their Championship opponents throughout, with only the post keeping Bristol level on two occasions.
Crawley finally broke the deadlock 17 minutes through leading goalscorer Matt Tubbs. Sergio Torres was the creator, crossing for Tubbs who struck his effort beyond former England international David James in goal.
The tie was put out of City's reach nine minutes from time when midfielder Stuart Pearson was handed his marching orders for a second bookable offence.
Crawly reached the fifth round of the competition last year and are now one round away from matching that feat again.