Stanley started brightly and took a deserved lead in the 15th minute when Matt Richards' attempted clearance fell to Nicky Hunt. His cross was nodded into the bottom right-hand corner of the net by George Bowerman, who was on trial with the Robins in pre-season.
Jermaine McGlashan forced Marcus Bettinelli into his first save in the 35th minute and the hosts were inches away from a leveller just before the break when Richards' cross-field pass was controlled by Jamie Cureton, who then thumped a shot against the crossbar from 18 yards out.
Accrington doubled their advantage almost immediately with Bowerman this time the provider, crossing for Luke Joyce to nod in from close range after 43 minutes.
Cheltenham did improve in the second half with Terry Gornell denied by the legs of Bettinelli and Kemar Roofe firing a shot over the bar over inside the six-yard box.
Cureton restored the Robins' hope in the 72nd minute with a vicious shot which cannoned in off the bar for his sixth goal of the season, but Stanley held on for a valuable three points.