It was comfortable for Eddie Howe's men, who had few problems in notching a club record-equalling seventh successive Football League win.
The Cherries made a flying start with Brett Pitman converting a fifth-minute penalty after James Hurst fouled Marc Pugh.
Pitman, League One's player of the month for March, has now scored in each of his last seven games and he took his tally for the season to 18 goal eight minutes into the second half when he curled home an unstoppable shot from 25 yards out.
Pitman was also involved in Bournemouth's third goal just after the hour mark when his near-post header from a corner was sliced into his own net by home defender Jermaine Grandison.
It was a tough afternoon for Shrewsbury, still not safe from relegation, in which chances were extremely hard to come by.
Skipper Matt Richards looped a header from James Hurst's cross against the crossbar early in the second half but it was a rare scare for the Cherries.
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