All the goals came in the first half, but there could have been more for either side as chances came and went throughout the contest.
Cheltenham scored their first win in five attempts over Bristol City since gaining admission to the Football League in 1999, while it's now four consecutive league and cup defeats for Gary Johnson's City team.
Both sides had great chances to open the scoring within the first ten minutes of a lively first half.
Cheltenham's Kayode Odejayi saw a header saved from a Brian Wilson cross; then Scott Murray of Bristol City accepted a pass from Enoch Showunmi and sidestepped a defender only to see his shot held by goalkeeper Shane Higgs.
Cheltenham took the lead on 24 minutes, when striker Steve Guinan rose highest to send a looping header beyond goalkeeper Adriano Basso from a Craig Armstrong cross.
The lead was doubled nine minutes later when Basso failed to get any real distance on his punch from another Armstrong cross. Wilson was waiting unmarked on the edge of the area and sent a volley crashing back into the net.
Basso then saved again from Odejayi and Guinan smacked a shot against the crossbar as Cheltenham threatened a third goal, although City hauled themselves back into it six minutes before the break when David Cotterill scored left-footed from the edge of the area.
Cheltenham should have extended their lead at the start of the second half when Basso, who had a nervous evening, spilled a Grant McCann free-kick to Guinan only two yards out. The striker thumped his shot at goal but it hit Basso's body and somehow bounced out.
City's Phil Jevons saw a low shot held by Higgs but Basso was beaten again on the hour mark when Odejayi's header flew past him but rebounded off the underside of the bar.
More work, then, for Bristol City to do on the training ground but Cheltenham can look forward to the second round for only the third time in their history.