The first half saw both teams trying to adapt to the sticky Home Park surface and did not give many chances away.
The Argyle supporters thought their side should have been awarded a penalty on the quarter-hour mark when Jamie Mackie was brought down in the box. The striker had tried to get on the end of Gallagher's pass but the referee was unconvinced that he had been felled by Nick Montgomery.
As half-time approached, Alan Judge fired his shot over and then, on 41 minutes, he danced in from the right and fired a left-footer against the foot of Blades goalkeeper Paddy Kenny's near post but, fortunately for the visitors, Rory Fallon was unable to scoop up the rebound.
Plymouth were the first to score as second-half play resumed courtesy of Fletcher. The debutant, on-loan from Crystal Palace, was on the end of Gary Sawyer's corner and blasted the ball in from a crowded penalty area into the top corner.
Argyle were not used to taking the lead and it showed when United drew level on the hour mark. Goalkeeper Romain Larrieu scuffed his goal-kick and Chris Barker was unable to keep control of the ball, allowing Webber to race clear and slip it past the custodian.
The Pilgrims fought hard and regained the lead after Sawyer's throw-in was tipped off by Fallon into the path of Gallagher. The striker, on-loan from Blackburn Rovers, picked his spot, curling the ball into the bottom corner, giving Kenny little chance.
Argyle's lead lasted all of two minutes after Gary Naysmith was handed a free-kick outside the box. His delivery found Halford unmarked and in a prime position to head past Larrieu, again in front of the Blades fans.