Richard Walker's sixth goal of the season was enough to separate the sides and decide a fiercely-contested encounter, that saw Darlington finish the match with ten men following the 84th-minute sending-off of midfielder Ricky Ravenhill.
Beginning the game outside the League Two play-off places for the first time this term, against a Darlington outfit whose recent good form had seen them climb up into third place, Shrewsbury made the ideal start - and ultimately settled the issue in the fifth minute.
When the Quakers defence failed to clear a Chris Humphrey centre deep into the goalmouth, the ball was returned into the box - and during the ensuing scramble Walker was on hand to force the ball home from six yards.
Despite the early setback the visitors looked for a leveller and it required some committed defending to keep them at bay - one block in particular from full-back Darren Moss, which thwarted a goalbound effort by Danny Carlton, highlighted Shrewsbury's determination to protect their lead.
Without 20-goal top scorer Grant Holt - who was serving the second of his two-match suspension - Shrewsbury found further chances harder to come by and, although central defender Graham Coughlin headed over an inswinging corner from Ben Davies, it was Darlington who reached the break still forcing the pace.
After the interval it required fine stops from Shrewsbury keeper Luke Daniels to deny efforts from Pawel Abbot and Steve Foster but, as the half progressed, the home side emerged to create some opportunities of their own - the best of which fell to substitute Dave Hibbert, who was unable to convert a Chris Humphrey centre.
Six minutes from time Ravenhill was ordered off for a second cautionable offence and Shrewsbury made it through safely to the final whistle - collecting three vital points and climbing above Darlington in the process.