Full-back Peter Cavanagh was tripped on the edge of the area by Rhoys Wiggins to concede the spot kick. But Accrington midfielder Andy Proctor blasted his penalty high over the bar into an adjoining car park.
Accrington were still reeling from this set-back when in the 86th minute Pitman volleyed in a centre from the right from Jake Thomson to earn Bournemouth three precious points.
Accrington's hero was their goalkeeper Kenny Arthur who made a string of outstanding saves starting with an early block from Thomson and then making three more in the second half to deny Pitman, Thomson and Steve Fletcher.
Danny Hollands struck the inside of an Accrington post with a superb volley from the edge of the area but Accrington always looked dangerous on the break.
Paul Mullin had a header turned over the bar and Jimmy Ryan also went close in an evenly contested battle.
Referee David Phillips was a busy man, booking Ryan, Proctor and James Bell of Accrington and Bournemouths' Thomson and Mark Molesley.
Pitman might have scored a hat-trick but snatched at two clear opportunities after being played clear on both occasions by powerful headers from big target-man Fletcher.
Fletcher was denied his first goal since re-joining the Club from Crawley with a header which Arthur somehow managed to keep out.
Accrington introduced Craig Lindfield, on loan from Liverpool, who has spent the first half of the season with Bournemouth but he had little chance to shine after taking over from John Miles in the 57th minute.