Rovers made a positive start to the afternoon, threatening the visitors' goal through Ryan Lowe before breaking the deadlock in the 19th minute.
Shrewsbury almost pulled one back in spectacular fashion at the half-hour mark, as Luke Summerfield spotted Rovers keeper Owain Fon Williams off his line and went for goal from the halfway line, missing the target by inches.
The Shrews applied some pressure ahead of the break - with Fon Williams saving from Tom Eaves in the 36th minute before Jon Taylor missed the target from inside the box - but were unable to level the tie before the referee brought the half to a close.
Rovers edged the early stages of the second period and almost doubled their advantage just after the hour mark when Ash Taylor headed a Power free kick against the bar and Matthew Pennington put the rebound wide.
The Shrews soon raised their game and put Tranmere under the cosh, drawing level in the 72nd minute as Paul Parry headed Eaves's free kick home from close range.
But it was Tranmere who dug deepest to secure the win, Brown beating Chris Weale from 18 yards after being picked out by a long ball from Power to hand his side a boost in their mission to stave off the drop.