Torquay midfielder Nicky Wroe grabbed a hat-trick as the hosts came from behind to beat their League Two rivals and reach the second round of the FA Cup.
Cheltenham took the lead when Theo Lewis headed Justin Richards' cross.
But Josh Low fouled Elliot Benyon in the box to allow Wroe to level from the spot while Lee Ridley conceded a second penalty - and saw red - for a handball.
Wroe once again dispatched the spot-kick and he went on to complete his treble on the break in the second half.
Torquay United manager Paul Buckle told BBC Radio Devon:
"The sole purpose of today was to try and go through and we've done that, and I think in the end we've done it with some real style.
"We fell behind and I was disappointed about that, but the players then stepped on and in the last half an hour we were exceptional in our attacking play."
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