With two minutes left the on-loan Reading man drilled into the corner to break the Pilgrims' hearts as Kevin Blackwell's side continued their recent upward momentum.
United came into the game as the form team following back-to-back wins over Bristol City and Peterborough, but it was Plymouth who made all the early running.
And had it not been for some early heroics from on-loan Carl Ikeme they would have gained reward for their industry as first Rory Fallon and then Ashley Barnes forced the goalkeeper into smart saves from 12 yards.
In between, Joe Mason drilled wide of the target as the Blades struggled to find their stride.
The absence of top scorer Darius Henderson, missing with a thigh complaint, clearly hampered the visitors, who looked woefully short of ideas going forward.
It was 41 minutes before they registered a shot on goal of any description, and even then Nick Montgomery's distance effort was blocked by Gary Sawyer.
Montgomery's frustration was compounded even further by a rash booking prior to the break as he was cautioned for a clumsy challenge on Alan Judge.
Whatever was said by Blades boss Blackwell at the interval, however, clearly did the trick and 10 minutes after the restart they came within the width of a post of taking the lead as Montgomery again took aim from distance, only to see his 25-yarder cannon back off the woodwork.
Fit again Kari Arnason should have done better for Plymouth as they also sought a breakthrough, but he scooped the ball over the crossbar when Judge's corner dropped to him in the box.
Finally, United were starting to assert some pressure and in an effort to capitalise they sent on Ched Evans for Lee Williamson with 24 minutes left.
At the other end, Judge curled an effort over the crossbar while Fallon - fresh from his recent heroics in firing New Zealand to the World Cup - continued to cause problems with his willingness to put himself about.
It was Carl Fletcher, though, who was left cursing his luck as his 25-yarder smacked back off the post with seven minutes left.
And five minutes later struggling Plymouth's plight was brought into sharp focus as Harper picked the ball up 25 yards out and drilled an angled shot past Romain Larrieu to snatch the points.