Hinton was quick to block an effort from Midgley as Hartlepool turned up the pressure and forced a string of corners from the home side.
The visiting fans and players also twice appealed desperately for penalties after handball claims against Harriers right-back Medou-Otye.
The determined Broughton did brilliantly on the left on 78 minutes and squared for sub Bird but the Welshman shot wide when he should have done better.
Hartlepool made a straight switch on 79 minutes as Sharp came on for Lee.
With only a minute of normal time left the 3,748 crowd was stunned as Sharp nipped in to tuck home Stephenson's free-kick from close range.
Harriers defenders and keeper were guilty of ball-watching for the last-gasp goal.
There was more woe for Harriers when Broughton was sent off for a second bookable offence.
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