Harrad, who joined from Bury earlier this month, grabbed the only goal after 73 minutes in a game that was delayed for 15 minutes during the first half by torrential rain and lightning.
Adam Mekki and Michael Rankine had earlier wasted the Shots' best chances and they were made to pay by Harrad as the Robins made it seven points from a possible nine.
Robins centre-half Steve Elliott went close to opening the scoring in the sixth minute when he reached a free-kick from Billy Jones, but Jamie Young parried his effort to safety.
The heavens opened and with thunder and lightning overhead the players were taken off the field in the 20th minute. Aldershot improved after the hold-up and Mekki shot wide from the edge of the box. Rankine mis-kicked from a good position and Scott Brown scooped away a header from Peter Vincenti as the first half ended goalless.
Harrad settled the contest, heading Jones' corner over the line at the near post and Cheltenham closed the game down well.