Bristol City will join Coventry City in the Johnstone's Paint Trophy Area Quarter-Finals draw on Saturday, after winning 3-1 at Cheltenham Town.
Burns put City ahead in the 28th minute and Korey Smith added a second in the 56th minute.
Cheltenham pulled one back through Gornell in the 64th minute but a stunner from Smith in the 84th minute, put Bristol City in the next round with the Sky Blues, Gillingham, Leyton Orient, AFC Wimbledon, Plymouth, Crawley and Northampton.