Exeter created the better chances and both Ryan Harley and Richard Logan were denied by the woodwork.
But a superb 25-yard strike from Murray midway through the second half clinched all three points for the visitors.
Shrews skipper Ben Davies tested the goalkeeper with a long-range effort in the second minute, but it was the home side who settled best in the difficult conditions.
Harley's fine cross picked out Marcus Stewart who volleyed into the sidenetting from eight yards.
Exeter created a host of chances throughout the first half and should have been ahead at the break.
Matt Gill struck the ball over the bar after Harley's corner was half cleared, before Harley's free-kick from 25 yards was superbly tipped over the bar by Luke Daniels.
At the other end Paul Jones kept out a close-range strike from Richard Walker with his feet, but it was Exeter who remained on top.
Adam Stansfield fired over the bar after he was picked out by Rob Edwards and Logan's header from Harley's corner was deflected on to the bar.
Harley also hit the woodwork with a powerful strike against the bar from the edge of the box with the goalkeeper beaten.
Exeter continued to dominate after the break. George Friend thumped the ball over the bar after a fine run, before Gill sliced a shot narrowly wide from the edge of the box.
Harley, who was outstanding all afternoon, struck the ball into the arms of the goalkeeper and Stansfield struck the ball wide from 15 yards.
But completely against the run of play Shrewsbury took the lead in the 71st minute. There seemed no danger when Murray controlled the ball on near the left touchline, but he shocked everyone with a strike from 25 yards which flew past Jones.
In the dying minutes Harley and Gill were both denied by Daniels, but there was no way back for the Grecians.