The Football League newcomers took the lead in the 32nd minute when David Ball struck from close range after Town goalkeeper Scott Brown parried Junior Brown's shot. The hosts had been on top with Jermaine McGlashan missing two clear chances to open the scoring.
Junior Brown was denied by Scott Brown in the 54th minute after skipping past Sido Jombati, and Steve Elliott cleared another Junior Brown effort off the line as Fleetwood looked capable of extending their advantage. But Cheltenham substitute Jeff Goulding levelled with his first touch, stroking home after Elliott's header from a Billy Jones corner was kicked off the line in the 64th minute.
Town were now on top and Jombati scored with a fine diving header from another Jones corner in the 75th minute.
Fleetwood substitute Jon Parkin hit a post from close range in the 81st minute after more fine play from Junior Brown and Steve Gillespie floated his follow-up effort inches over the bar.
But Brown prodded in from close range deep into stoppage time and Cheltenham skipper Alan Bennett was then sent off for violent conduct.