Darlington have been rocked by a virus that had swept through the club over Christmas, but they made five changes from the side that lost to Mansfield on Boxing Day.
The visitors made the brightest start to the game and, in the fifth minute, midfielder Ben Davies went close with a 35-yard free kick.
The hosts, however, should have taken the lead a minute later when defender Patrick Collins met a Darren Holloway free kick at the far post. Collins directed his header into the ground and the ball bounced over the crossbar from only four yards out.
Shrewsbury had a goal disallowed in the 15th minute when striker Andy Cooke fired home. The striker was offside and the goal was chalked off.
The visitors took the lead in the dying seconds of the half and goalkeeper David Stockdale should take some of the blame. The keeper stayed on his goal line instead of collecting a Neil Ashton cross and when the ball fell for Michael Symes he despatched his header into the net from close range.
The home side needed to respond early in the second half and they did so just three minutes after the break. Julian Joachim collected a header from Michael Cummins and fired a low drive from 12 yards under Shrews keeper Scott Shearer.
As the game wore on a draw looked to be on the cards, although Shrewsbury broke away in the 68th minute through Cooke who sent a right-wing cross for Symes. The striker turned quickly but, his shot from inside the area was straight at Stockdale.
But the visitors kept moving forward and they got their reward three minutes from time when Davies sent over a right-wing cross and Stockdale again failed to collect leaving David Edwards a simple header from close range that proved to be decisive.