Pitman struck for the fourth time in as many games and the eighth time this season as his inswinging corner fooled Shrewsbury keeper Luke Daniels, who was caught in no-man's land and could only watch the ball sail over his head.
Town's best first-half chance fell to Ben Davies, who picked up a loose pass from midfielder Marvin Bartley to fizz a shot inches over the bar from 25 yards.
Bournemouth were good value for their victory although they ground out a result on the back of some determined defending.
Pitman wasted two opportunities to add to his tally at the start of the second half, first firing over from 14 yards after cleverly working himself a yard of space and then making a hash of his effort after on-loan Southampton winger Jake Thomson picked him out with his pin-point cross.
Those misses proved not to be costly as Bournemouth battled determinedly to hang on to their lead.
Central defenders Shaun Cooper and Jason Pearce were outstanding in keeping 21-goal marksman Grant Holt and Nick Chadwick quiet as Shrewsbury piled on the pressure in the second half.
But for all their territorial advantage and possession the Shrews failed to draw a save from Bournemouth keeper Shwan Jalal.
The nearest Shrewsbury came to snatching a point was when defender Darren Moss met a near-post corner and looped a header on to the top of the Bournemouth bar.
To rub salt into the wound Moss received a straight red card from referee Richard Beeby in injury time for kicking out at Pitman.