The Cherries earned a fifth win in six matches since manager Eddie Howe returned to the club.
Charlie Daniels opened the scoring when he ripped a low third-minute free-kick through Shrewsbury's defensive wall after Darren Jones had handled on the edge of the penalty box.
Three minutes later Marc Pugh cut in from the left to tee-up a simple header for Josh McQuoid, the striker's third goal in two matches.
Shrewsbury clawed a goal back through Matt Richards from an 11th-minute penalty awarded for a soft Miles Addison challenge on Marvin Morgan.
But the visitors, with only one win in their previous 10 matches, were fortunate to still be in the contest at half-time as Bournemouth squandered a catalogue of chances, with Lewis Grabban twice a guilty party and Simon Francis denied by goalkeeper Chris Weale.
Bournemouth continued to dictate the play after the break, but Shrewsbury were more compact and assured in defence and restricted the Cherries to a series of half-chances.