Against the run of play it was the hosts who had the first big chance of the game only for Byron Harrison to blast over from close range after Cheltenham keeper Scott Brown failed to hold a Luke Moore cross.
Wimbledon were made to pay for that miss after 25 minutes as Cheltenham's tidy passing led to the opening goal when Marlon Pack met a stray header from Will Antwi with a ferocious half-volley outside the area, finding the bottom corner of Seb Brown's goal.
The Dons reacted positively to going behind, and came close when Moore's audacious flick found Jack Midson, whose looping header rebounded off the crossbar.
Despite the Dons' resurgence, Cheltenham doubled their lead after 56 minutes. Former Wimbledon player Kaid Mohamed scoring with a close-range header after Billy Jones' cross was flicked on by an unmarked Steve Elliott.
Wimbledon again rallied and pulled one back in injury time as Harrison headed home Jim Fenlon's superb cross, but it proved to be just a consolation for the hosts.