With the seconds ticking away into the two minutes added on at the end of the match, Henderson had plenty of time to clear an innocuous pass upfield, but he slipped on the greasy surface and Pitman had time to gain possession and shoot into an empty net.
Brighton had no time to get back into the match and suffered their second defeat in three days.
For Bournemouth's beleaguered manager Kevin Bond it was only his second win in the league since he took charge in October.
The mistake was rough justice on Irish International goalkeeper Henderson who had produced a competent display to keep out first-half shots from Darren Anderton and Simon Gillett.
But his best save came in the 71st minute when he somehow turned a header from Sam Vokes against the post and away from danger.
Goal hero Pitman should have scored with ten minutes remaining but mis-hit a shot from 15 yards with only the goalkeeper to beat.
Gillett, who returns to Southampton at the end of a loan period, went close with two shots through the crowded goalmouth as Bournemouth struggled to turn their pressure into goals.
Alex Revell headed into the side netting for Brighton and Cherries goalkeeper Neil Moss saved with his legs a shot from Joe Gatting.
The match looked to be heading for a goal-less draw until Pitman struck with his fourth goal of the season to give Bournemouth hope of avoiding relegation.
Referee Richard Beeby booked Bournemouth's Marcus Browning and Revell for a midfield flare-up and a minute later showed a yellow card to Seagulls captain Dean Hammond for a foul.
Brighton appealed for a penalty when Jake Robinson appeared to be pushed in the 57th minute as he competed for a centre from Sam Rents, but Brighton posed little threat as an attacking force.
Bournemouth had plenty of possession in the second half but until Henderson's blunder looked like being denied only their fifth league win of the season.