The second-placed Cherries, who have won only three times at home in League Two since mid-October, went into the game buoyant after back-to-back away wins at Crewe and Rotherham. But they seldom looked capable of unlocking third-from-bottom Cheltenham, who produced a resilient display.
Only in the frenetic closing stages, when Brett Pitman saw a free-kick beaten out by goalkeeper Scott Brown and Cheltenham defended in numbers to soak up sustained pressure did Bournemouth show any real cutting edge.
Cheltenham had only managed one win in their last eight outings, but did enough to warrant a share of the spoils.
Their best chances came early on when Bournemouth goalkeeper Shwan Jalal beat out Justin Richards' low shot with his legs and Michael Pook saw a speculative lob tipped over the bar.
Bournemouth huffed and puffed without much menace and their prospects of winning ended when Pitman fluffed a late chance and then saw his free-kick saved.