Goals from Steve Elliott and Darren Carter secured the win but the Robins had to endure a late onslaught after Dons substitute Brennan Dickenson rifled in an effort with his first touch.
Cheltenham almost took the lead after eight minutes when Sam Deering turned a Billy Jones cross into the path of Carter and his low curling shot forced veteran keeper Neil Sullivan into an excellent diving save.
The opener came two minutes later when Jones curled a brilliant free-kick into the six yard box for the incoming Elliott to direct a header into the bottom right-hand corner.
Carter doubled the Cheltenham lead after 57 minutes, rounding off a flowing move with a well-placed effort into the bottom left of the goal.
Dickenson's 61st minute effort handed the visitors a lifeline and they should have levelled 15 minutes from time when the substitute took the ball past goalkeeper Scott Brown before picking out Jack Midson, but his shot sailed over the crossbar.
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