The game look destined to end 0-0 before Luke Wanadio drilled home from inside the area in the 91st minute.
However, the Robins, who had lost for the first time in 22 games in midweek, forced an equaliser when Dan Holman scored with the last kick of the match.
Both teams had hit the woodwork earlier with Wanadio hitting the post and Jake Munns doing the same for Cheltenham.
Cheltenham Town boss Gary Johnson told BBC Radio Gloucestershire:
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"My dominant emotion is one of disappointment of course, you've got to give the opposition credit today - they took advantage of us not quite playing well enough and it just gave them a bit of confidence.
"I thought five or six of our boys weren't on it today, I could see in their faces and in their game that they were having a panic and couldn't control the ball.
"There's nine games to go and we're still in the driving seat, two points clear in the league so we've got to make sure we keep pushing the boys along and help the ones that look like they've got the pressure on their back a bit."
National League leaders Cheltenham salvaged a last-minute point having conceded in injury-time at Welling.