The Cherries deserved their first away league win of the season thanks to a battling second-half performance.
The result also gives them hope of being able to climb out of the bottom four by the end of the season.
But the 52nd minute winner from James Hayter was thanks to some slack defending from the Spireites.
A left-wing corner from Darren Anderton was headed across goal by Sam Vokes. Chesterfield failed to clear the ball and Hayter was able to slide his shot into the net for his first strike in ten games.
Chesterfield, with two key players suspended, only really got going after falling behind.
The 57th minute introduction of midweek signing Jamie Ward gave them a boost, but despite plenty of pressure they could not break the visiting defence.
Paul Shaw went close in the 60th minute with a low shot from the edge of the box which brought a diving save from Bournemouth keeper Neil Moss.
Sixty seconds later Moss made a superb one-handed save to push over a 25-yard shot from Peter Holmes.
Chesterfield kept up the pressure and Ward broke into the box only to see his shot blocked by Marcus Browning.
Then Holmes got clear down the right and sent over an excellent cross only for Shaun Cooper to nick the ball away as Paul Hall waited to score.
Bournemouth continued to look dangerous on the counter attack and teenager Jack Cork smashed an effort over the bar.
But they went close to scoring a second goal nine minutes from the end. Anderton forced his way down the right flank before sending over a cross which was deflected into the path of Hayter. The striker was left with only the keeper to beat from the edge of the six-yard box but his effort lacked power and Michael Jordan was able to save.
The action in the second half was in contrast to a dour first 45 minutes. The only chance fell to Holmes who cut in from the left before sending in a shot which Moss did well to keep out.