Midfielder Danny Rose had handed the visitors a first-half lead, but a looping Paul Benson header hauled the Robins level.
The home side had the better of the opening exchanges and Michael Hector crashed a 25-yard effort against the crossbar.
But the breakthrough came from Aldershot after 24 minutes, when the ball was worked to Rose, who threaded his shot through a crowded area.
Kieron Cadogan and Jeff Goulding had efforts for the visitors before the break, but the second half belonged to Cheltenham.
Byron Harrison was denied by Shots keeper Jamie Young, who then pulled off a brilliant diving save to keep out a Kaid Mohamed effort.
The 67th-minute equaliser followed a Keith Lowe long throw which was helped on to Benson, who looped his header over the goalkeeper.
Any hopes the visitors had of finding a winner were dashed 17 minutes from time, when defender Anthony Tonkin picked up a second yellow card.
