Jonjo Dickman hit the all important 80th minute winner, after sub Neil Wainwright had cancelled out Adam Connolly's first-half opener for the visitors.
Guylain Ndumbu-Nsungu completed an amazing turn around when he added a third in injury time.
Cheltenham should have taken a ninth minute lead through Damian Spencer. After a well worked free-kick, Grant McCann picked out Brian Wilson on the right who crossed to Spencer, but the forward could only direct his header wide of Sam Russell's goal.
Darlington went a goal down on the half-hour mark. Connolly's initial shot was blocked by Shelton Martis, before the ball dropped back out to the Robins midfielder who unleashed an unstoppable drive past Russell.
After an ineffective first half, Johnson came agonisingly close to scoring for Darlington when he curled his efforts inches wide.
With 14 minutes left Quakers finally drew level, through Wainwright, when he jumped highest in the box to send his header past Shane Higgs from Dickman corner.
After Wainwright's equaliser the floodgates opened for the home side. Latching onto Anthony Peacock's through ball Dickman scored his first goal of the season, slotting past Robins keeper Higgs.
Ndumbu-Nsungu made sure of the victory, acrobatically turning his shot past Higgs.