Stewart came charging out of his area on 62 minutes to try and intercept a Paul Shaw throughball only to misjudge his clearance and allow Colin Larkin an open route for goal.
Earlier Derek Niven had given the away side the lead just after half-time and the win was polished off 30 seconds from time by Paul Shaw.
The Cherries miserable afternoon began as early as the sixth minute when captain Marcus Browning limped off with a calf injury and things rarely improved from there.
Larkin should have scored on five minutes when he attacked Shaw's low cross only to fire his effort well over from 15 yards.
Steven Foley-Sheridan had a penalty appeal for the home side turned down on 15 minutes when Reuben Hazell appeared to handle his cross and although the assistant referee temporarily raised his flag, referee Paul Taylor ignored the appeals.
Chesterfield continued to press in the first half and went close through Mark Allott and then Paul Hall but the first half petered out.
Niven's opener two minutes into the second period came after Kevan Hurst crossed from the left-hand side, stand-in skipper Karl Broadhurst struggled to clear and the Scottish midfielder finished well from just outside the penalty area.
Bournemouth's top scorer last season, James Hayter then missed what proved to be their best chance of the game.
Hayter sprayed the ball out wide to loanee Lionel Ainsworth on the right-hand side who cut back onto his left foot and crossed for the Cherries' striker to head narrowly wide from six yards.
Twelve minutes later came Stewart's moment of madness.
As Chesterfield broke Shaw delivered a perfectly weighted pass for Larkin to chase onto, but Stewart looked favourite to win the ball.
However, his attempted tackle only allowed the Chesterfield midfielder arguably the easiest goal of his career.
Bournemouth manager Sean O'Driscoll threw on club captain Steve Fletcher with 15 minutes to go but it was Chesterfield who continuted to look the greater attacking threat.
And Shaw got himself on the scoresheet with a well taken finish just inside the penalty area.