But brave Cheltenham made a good case for at least a share of the points with a brave attacking display which saw them waste several clear chances to equalise a second time.
Best could have had a hat-trick but hit the bar and was unable to convert two other opportunities, but his goal gave Bournemouth something to celebrate after a disappointing start to the new season.
Bournemouth were ahead in the first minute when Callum Hart crossed from the left and the vastly experienced Steve Fletcher in his 14th season with the club, scored with the outside of his right foot, a shot which goalkeeper Shane Higgs reached but could not keep out.
Cheltenham were level in the 14th minute, soon after Best had been unable to tuck away a shot from six yards.
Cheltenham's goal came from Northern Ireland International Grant McCann, who ran at the retreating Bournemouth defence before unleashing a left-foot drive from fully 35 yards which hit the inside of the post before going in.
Cheltenham may have taken the lead in the 31st minute when Kayode Odejayi got clear but Gareth Stewart was quickly off his line to block the shot.
Cheltenham went close again in the 50th minute when Hart cleared off the line from Steve Guinan.
But Bournemouth, playing with confidence and attacking zest, should have scored after 53 minutes only for Best to smash a close range shot against the bar when it looked easier to score.
Best made up for his error almost immediately when Steven Foley crossed from the left and he got in front of two defenders to head home from ten yards.
Best was clear again on the hour as he wriggled free of the defence only for Higgs to quickly emerge from his line to smother the shot.
Instead of going on to win handsomely, Bournemouth retreated into their defensive shell and almost paid for it as Cheltenham missed several clear openings.
In the 72nd minute Guinan hit the bar with a header and as John Finnigan prodded the rebound towards an empty net, Stewart came from behind his goal line to save.
McCann had a great chance in the 77th minute to score his second equaliser but his free kick from 25 yards was pushed to safety by Stewart diving to his right in spectacular fashion.