Second-half goals from teenage midfielder David Edwards – who also scored in the defeat of Darlington – and man of the match, John Grant, enabled the Shrews to record back-to-back victories for the first time this term and continued their climb away from the lower reaches of League Two.
Edwards struck on 64 minutes and Grant six minutes later to ultimately a secure a thoroughly deserved success for Gary Peters' men who dominated this highly entertaining encounter – played on a treacherous playing surface from the early stages.
First blood, however, was so nearly drawn by the visitors with Shrewsbury thankful for a superb save from goalkeeper Scott Howie to thwart a second minute snapshot from Rovers defender Dave Savage.
But as the half unfolded the home began to hold sway with Rovers goalkeeper Ryan Clarke saving well from Jamie Tolley, although on the stroke of half-time Clarke could only watch as Jake Sedgemore's free-kick smashed into the crossbar above his head.
After the break Rovers also hit the woodwork when Lee Thorpe struck a post from a tight angle.
At the other end, Kelvin Langmead and Sam Aiston went close to making a breakthrough for Shrewsbury, but just a minute after Clarke had saved well from Tolley again, Edwards latched on to a through ball from Grant to score from 12 yards.
From being the provider, Grant then turned scorer rising to head home a Sedgemore centre from eight yards to put the outcome beyond any doubt.
To complete a bad day for Rovers boss Ian Atkins – a former Shrewsbury favourite in his playing days – defender John Anderson was given his marching orders 16 minutes from time following a clash with Shrewsbury striker Langmead.