Clarke kept Harriers in the game on 74 minutes when Beckett gave to off side trap and raced onto Breckin's through-ball only for the big keeper to save well at his feet.
Meanwhile, Harriers were not giving the home side any cause for concern.
The Harriers keeper easily held a header from Edwards 12 minutes from time.
The visitors were back by a vociferous travelling support from Worcestershire.
Chesterfield replaced forward Reeves with former Birmingham City player Willis on 79 minutes.
Edwards was booked for Chesterfield on 82 minutes for holding back Skovbjerg on the right-wing. Thomas Skovbjerg
Irish midfielder Doyle came on in the 84th minute in place of Webb.
Hadley became the third Harriers player to be booked on 85 minutes for a clumsy challenge.
Harriers had their first decent shot on target a minute from time after Smith won a free kick 20 yards from goal.
Ducros curled the free kick towards goal and it took a fingertip save from Pollitt to keep the ball out of his net.
Chesterfield came close to a second goal in injury time but no-one was on the end of Willis's dangerous low cross.
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