Harper's 88th minute drive following a late corner, which the United midfielder had himself won by forcing home goalkeeper Romain Larrieu into a fine save, was enough to separate two well-matched sides.
United had been dealt a pre-match blow when in-form striker Darius Henderson, whose hat-trick seven days earlier had earned a 3-2 victory at Bristol City, was ruled out with a thigh injury.
Richard Cresswell came in, despite requiring strapping on a rib injury.
The Pilgrims, who had survived a pasting the previous weekend when their home game with Barnsley was abandoned with the visitors 4-1 up, gave a debut to teenage forward Joe Mason and made three other changes.
First-half chances were few and far between. An early header by Argyle's New Zealand international forward Rory Fallon that United goalkeeper Carl Ikeme held easily was as close as either side went to breaking the deadlock.
Republic of Ireland under-21 captain Alan Judge was prominent in much of Argyle's work and one burst from central midfield threatened to inflict damage on the Blades before a well-timed tackle from captain Chris Morgan stopped him in his tracks.
Argyle came out of the blocks strongly at the beginning of the second half and central defender Kari Arnason headed narrowly over from a corner.
United responded strongly and came within inches of breaking the deadlock when Argyle failed to clear their lines and Nick Montgomery's long-range drive crashed against the post.
Man-of-the-match Judge responded with a delicate chip form distance that dropped on to the top of the goal, with Ikeme damaging his back as he stretched to cover the shot.
Hampered by Ikeme's injury - and with no substitute goalkeeper on the bench - United kept the Pilgrims at bay, with captain Carl Fletcher hitting the post in the final minutes, before Harper settled the issue.