The former AFC Wimbledon forward drilled a 20-yard shot just inside the post seven minutes from the end of extra time.
In a scintillating tie, Gary Alexander headed the visitors into a 23rd-minute lead, only for Matt Richards to fire home the equaliser from the penalty spot five minutes before the break.
Two goals from midfield maestro Nicky Adams then looked to have guided Sky Bet League One side Crawley into the next round.
The former Leicester and Rochdale playmaker diverted a clearance into the Cheltenham net on the hour-mark and then coolly wrong-footed home goalkeeper Scott Brown five minutes later.
But Mark Yates' outfit came back in style. In the 70th minute the excellent Terry Gornell robbed Crawley defender Kyle McFadzean and found the net at the second attempt after his initial shot had been saved by Paul Jones.
And the Robins forced the tie into extra time when Harrison rose to head home his first goal from a corner five minutes later.