They were gifted with an early own goal, but then defended well in a first half that was hectic at times for their defence.
But in the second half they had the measure of the home team, who ran out of ideas as the game wore on, and the visitors made sure with two late goals.
Shrewsbury cashed in on a gift goal from the home team, taking the lead on ten minutes after a quiet start to the game by both sides.
Striker Jon Daly was trying to help cover a right-wing free-kick from Ben Davies, which the defence failed to clear, and he sliced his effort to get the ball away into the roof of his own net.
Shrewsbury almost made it two on 19 minutes, but Gary Liddle's tackle was just enough to take the ball off the feet of Andrew Cooke after a great run and low cross from the right by the busy Derek Asamoah.
But for the rest of the half, it was all the home team. Willie Boland's half volley was well saved by Shrews keeper Chris MacKenzie, who also did well to keep out a low shot from James Brown.
But he was beaten by another Brown shot, only for the ball to be deflected high over the bar by Kelvin Langmead.
MacKenzie punched out another shot from Brown on 35 minutes, and Neil Ashton was in the right place at the right time to clear Daly's shot from the rebound off the line.
After the break, the home team's attacking output began to sag, and though MacKenzie did well to save a Daly shot as he dived to his right, that was the end of the day's serious work for the keeper.
The game began to move more to the other end, and on 77 minutes Shrews made it count when Stuart Drummond got on the end of another Davies free-kick and sent his long-range looping header into the roof of the net, with keeper Dimi Konstantopoulos rooted.
They added insult to injury on 82 minutes when Ben Herd got away on the right and crossed to the middle, where substitute Michael Symes was amazingly unmarked as he picked his spot with an easy close-range header.