Burgess went closest to opening the scoring 7 minutes into the second half. Stuart Elliott's cross fell to the targetman, but his effort hit Cheltenham keeper Steve Book's woodwork.
But Burgess set-up his strike partner Allsopp five minutes later to put City in front with a right-footed shot past the despairing Book.
Two minutes later, Burgess doubled the lead as Allsopp and Elliott counter attacked. Damien Spencer - who has scored four times already this season against City - was then shown a red card for elbowing Marc Joseph during the build-up.
The ten men were unable to reply, so City comfortably won the match 2-0.
City line-up (4-4-2): Myhill; Hinds, Joseph, Delaney, Dawson (Holt 45); France, Green, Ashbee, Elliott; Allsopp (Marshall 77), Burgess.
Un-used subs: Musselwhite, Melton, Forrester.