Joe Jacobson's goal sent the Shrews into half-time ahead before substitute Liam McAlinden sealed all three points for the hosts late on.
Shrewsbury almost raced into a third-minute lead when Cristian Lopez latched onto a long clearance from goalkeeper Chris Weale, but he headed narrowly wide with only visiting keeper George Long to beat.
United were restricted to efforts from long distance with Michael Doyle, Lyle Taylor and Marlon King all off target.
It looked like it would take something special to break the deadlock and left-back Jacobson duly obliged in the 39th minute with a brilliant 30-yard shot into the top corner of the net after the ball ran into his path.
Clough sent on Febian Brandy at the break but he was sent off just 12 minutes later for shoving Shrews sub Connor Goldson in the face.
Shrewsbury made their extra man count and added a second goal 10 minutes from time when Goldson burst forward and played in McAlinden, who took a touch before smashing a 10-yard shot to the roof of the net to seal a third win of the season for Graham Turner's side.