The Robins have now won their last three games and have now lost only twice in 11 starts.
Bournemouth, meanwhile, slipped back into the bottom four as their recent revival of four wins from six came to a halt.
Both sides had chances during a tightly contested first half and both goalkeepers were called into action within the first ten minutes.
Neil Moss of Bournemouth tipped a header from Cheltenham captain John Finnigan over the crossbar then, at the other end, Shane Higgs stretched to save from a Steve Fletcher header.
Higgs got down to hold shots from Josh Gowling and Marc Wilson, but as the first half wore on the number of chances for the home side gradually increased.
Former Liverpool youngster Steven Gillespie looked sharp in attack and he forced two saves from Moss.
Then Irish winger John Melligan drew a good save from the Bournemouth number one and produced a good run and cross from Damian Spencer to lift a shot over the bar.
After the break the tension among the players and in the stands was palpable. Both sets of players were frightened of making a mistake and the football became ragged as a result.
Possession was given away too easily and forward players at both ends of the pitch were forced to chase around looking for scraps to recycle into something in front of goal.
Bournemouth had the better chances of the second half with Fletcher sending a header just over the bar and James Hayter going close with a header and a curling left-foot shot.
A goalless draw looked a virtual certainty until Cheltenham won a corner late in the game with Finnigan's good solo run ending with a shot that was deflected behind.
Sosthene Yao took the flag-kick and although his first effort was cleared, the substitute found the space to cross again and this time Spencer charged in to power a header past Moss and clinch all three points.