The Shrews striker's 56th minute strike was enough to give Gary Peters' side a hard-fought win in the North-East.
The win for Shrewsbury continues their recent improvement in form following the midweek draw at Carlisle. But, for Darlington their woes show no signs of easing after they slipped to their second consecutive defeat, extending their run without a win to four games.
There was only two minutes played when Simon Johnson sent a glancing header wide of Joe Hart's goal from Carlos Logan's corner. And, after Johnson was fouled on the edge of the Shrewsbury area, Dutch defender Shelton Martis could only send the resulting corner over the bar.
At the other end skipper Matt Clarke produced two important blocks. First, charging down Gavin Cowan's effort, before denying the same player again on the rebound.
Quakers' on-loan striker Kyle Lafferty should have done better in the tenth minute when he blazed high and wide after taking on Ben Herd.
But, the best chance of the half fell to Quakers' other loan forward, Andy Cooke. After pouncing on a Richard Hope slip, Cooke, with only the goalkeeper to beat, scuffed his shot horribly across the face of goal.
There was a let off for Quakers midway through the first half when Herd could only fire over the bar after Clarke failed to deal with Neil Ashton's cross.
As the game began to fizzle out with half-time approaching Ashton tested Quakers keeper Sam Russell with a speculative long-range effort, which failed to cause any problems.
But, it was Kelvin Langmead who should have gave the visitors the lead when he made a hash of his shot with only Russell to beat. And Cooke was guilty of missing another good opportunity to score, heading Ryan Valentine's cross wide.
Quakers were made to pay for their profligacy in front of goal when McMenamin gave the Shrews a 56th minute lead. Mark Stallard saw his lob come back off the bar and McMenamin was the grateful recipient of the rebound, tapping home from close range.
The home side may have fallen further behind, but for superbly timed challenge from Martis to deny Langmead.
Cooke was superbly denied at the death when goalkeeper Hart turned his header wide as Shrewsbury hung on to claim the points.