United, previously without a home win since mid-February, dispensed with the services of Danny Wilson in midweek and tasked their former centre-half with the job of trying to steer them back into the Championship.
On this evidence, it appears an inspired move as Morgan's men were well worth their victory.
Chris Porter's 38th-minute effort ensured the Blades' superb first-half display did not go unrewarded. The striker, starting for the first time since returning from a loan spell at Shrewsbury, powered home Tony McMahon's in-swinging free-kick to end the hosts two month wait for a goal at Bramall Lane.
McMahon, who struck the woodwork with another set-piece soon afterwards, had earlier delivered the cross from which Harry Maguire had a shot deflected onto a post while Jamie Murphy twice went close as Swindon looked like being overwhelmed.
The visitors, though, began the second period in much more decisive fashion. Nevertheless, Porter could have doubled United's advantage on the hour mark after meeting Ryan Flynn's pass only for his control to let him down.
Simon Ferry produced a well-timed tackle to thwart David Kitson after Michael Doyle's pass fortuitously wriggled its way though.
But the former Reading centre-forward was not to be denied and made sure of the points with a commanding header from another superb McMahon delivery.