The home side should have taken the lead in the 11th minute when Joe Morrell played in Mo Eisa, but he blazed over the crossbar.
Accrington opened the scoring in the 22nd minute when Sean McConville’s corner was headed into the net by an unmarked Billy Kee for his 16th goal of the season.
A free-kick from McConville was headed over by Kee, who found space in the box again in the 35th minute.
Cheltenham debutant Elijah Adebayo, on loan from Fulham, fired over during first-half stoppage-time.
Accrington doubled their advantage in the 63rd minute when a long clearance from goalkeeper Aaron Chapman was not dealt with and Kayden Jackson slid the ball past Scott Flinders.
Eisa and substitute Brian Graham both saw shots cannon back off the bar in the 81st minute during one goalmouth scramble.
Janoi Donacien cleared off the line to deny Harry Pell in time added on as Cheltenham failed to find a breakthrough.
Gareth Bale wants first-team football and will discuss his future with Real Madrid this summer, his agent has confirmed.
Preston are reportedly weighing up whether or not to make a fourth offer for Peterborough United left-back Andrew Hughes.