The visitors got off to a quick start and created a few half chances before the Cherries finally got a grip on the game. Pitman got his fourth in four directly from a corner as the Shrews keeper flapped.
On the stroke of half-time Fletcher look set to burst through before being stopped in his tracks with Jackson getting a yellow for his troubles.
Early in the second half Pitman made space but managed to miss when it looked like he had to score. The visitors started to assert themselves and created a succession of chances but failed to put one away.
Moss got sent off in injury time and the Cherries got three points.