Cheltenham thought they had the match and a place in the FA Cup fourth round sewn up when they took a 2-0 lead with 16 minutes to go from their second penalty of the game.
But a looping header from Marcus Richardson and Drummond's late effort means that the teams must go again on January 17.
Goalscoring chances were at a premium during a tightly contested first half, although it was the home side that went closest to opening the scoring.
Midfielder and leading scorer Brian Wilson saw a shot deflected wide and sent another just over the bar, while defender Jerry Gill let fly with a goalbound long-range shot that had the sting taken out of it by a defender.
The best chance of the first half arrived on 28 minutes when Robins striker Kayode Odejayi shook himself free of his marker and bore down on goal with only debutant goalkeeper Stephane Gillet to beat.
The Luxembourg international sprinted off his line and made a brave block at the striker's feet to keep the scores level.
Both Odejayi and midfielder David Bird saw shots blocked during a goalmouth scramble at the start of the second half, but Cheltenham finally grabbed the lead with 58 minutes played when JJ Melligan scored from the penalty spot after Bird had been fouled.
A handball by Ben Davies gave Cheltenham their second penalty on 74 minutes, captain John Finnigan converting from the spot, but things began to go the way of the visitors in the final ten minutes.