That made it three wins on the trot for Keith Downing's side and they were full value for this latest success, as they battle to climb away from the danger zone.
Gillespie broke away to score their first on 11 minutes after a long clearance caught in the swirling wind, sliding his close-range shot under home keeper Arran Lee-Barrett.
Town dominated and thought they had a second when Steve Brooker fired home, but an offside flag saved the home team from further embarrassment.
But they kept on pushing forward and Alex Russell hammered a low shot against the post, before Brooker wasted a good chance by firing into the sidenetting and Gillespie's header was cleared off the line by Robbie Elliott.
After the break, Hartlepool woke up and took command themselves, but their finishing wasn't good either, as Richie Barker sent a header and a shot just wide of the target when goals looked certain.
James Brown had a shot cleared off the line by Alan Wright and defender Jerry Gill headed his attempted clearance against his own bar.
Sub Joel Porter went close with his first touch late in the game, but Town keeper Shane Higgs pushed his shot round the post for a corner and that set Cheltenham up for a grandstand finish.
They broke out of defence with a sweeping move down the right and as the ball was worked across the face of the penalty area and with the home defence stretched, it finished up at Gillespie's feet wide on the left and he finished with a great chip into the far corner.
With nine points from their last three games, Cheltenham are motoring, but Hartlepool's mid-season slump is beginning to turn into a crisis as they tumble down the League One table.