The Blades lacked a cutting edge for most of the game at Home Park but were thankful for a piece of magic from Harper.
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 momentum.
On-loan Carl Ikeme denied Plymouth’s Rory Fallon and Ashley Barnes early on with smart saves from 12 yards.
The Blades did come close to opening the scoring 10 minutes after the break when Nick Montgomery’s effort cannoned back off the woodwork and the woodwork then came to the rescue of the visitors as Carl Fletcher was denied from long range.
But with time running out Harper picked up the ball 25 yards out and drilled home.