Pitman's inswinging delivery after 20 minutes completely deceived Shrews goalkeeper Luke Daniels who could only watch as the ball sailed over his head and directly into the net.
Pitman should have added to his fourth goal in as many games and eighth of the season but missed two good chances at the start of the second half.
Firstly, he worked himself a yard of space only to blast over from 14 yards and he then failed to make good enough contact at the far post after he was picked out by Jake Thomson's pinpoint cross.
But despite exerting second-half pressure Shrewsbury never forced a save out of Bournemouth keeper Shwan Jalal, with a Darren Moss header against the bar the nearest Paul Simpson's side came to snatching a point.
To add insult to injury referee Richard Beeby showed Moss a straight red card in injury time for kicking out at Pitman.