But the Robins had to endure an anxious final few minutes after Rene Howe scored to give the Gulls late hope.
The hosts should have been ahead after just three minutes when new signing Paul Benson turned a Jermaine McGlashan cross goalwards only for the offside Harrad to touch it over the line.
Defender Michael Hector had a close range effort kept out by the legs of Torquay goalkeeper Michael Poke before Keith Lowe headed wide from close range.
The Gulls did manage to force home keeper Scott Brown into action midway through the first half when he turned a Howe shot away before Cheltenham scored after 33 minutes when Lowe headed home Kaid Mohamed's corner from six yards out.
The second half was just as one-sided with Harrad doubling the lead with a neat 12-yard finish in the 51st minute.
Russell Penn and Harrad were then denied by excellent saves from Poke before Howe hit his consolation goal two minutes from time.
