Goals in the opening minutes of either half, by Graham Coughlan and captain Paul Wotton set the Pilgrims on their way, with substitute Dexter Blackstock rounding things off with a late third.
Nothing went the way of United, who drew 1-1 with Arsenal at Highbury in the FA Cup fourth round on Saturday. Goalkeeper Paddy Kenny limped out of action with a hip injury and, with no substitute goalkeeper on the bench, midfielder Phil Jagielka took over between the posts.
Warnock will now recall keeper Phil Barnes from his loan spell at Torquay, but, although Kenny might be fit for next week's cup replay, midfielder Michael Tonge will definitely miss the match after picking a up a fifth booking of the season and a suspension.
The game was barely into its third minute when Coughlan set Argyle on their way to a first victory in ten games, sweeping home the ball after Sheffield United had failed to clear Tony Capaldi's long throw-in.
Argyle needed to defend for long periods of the remainder of the first half, but gave themselves breathing space 62 seconds after the interval when Wotton slammed home a long-range drive to end another move begun by Capaldi.
The Northern Ireland international completed a hat-trick of assists with two minutes remaining, when he set up Blackstock, the on-loan Southampton striker who had come on for Micky Evans, for a low diving header at the near post.
Captain Jagielka took over between the posts, a position he filled with some distinction for 45 minutes at Millwall earlier in the season when Kenny was sent off, with Paul Thirlwell brought into the midfield.
Argyle needed to test the stand-in, but found themselves pegged back by a Blades side who realised the best means of defence was attack.
The Pilgrims were up for the fight, none more so than Wotton, whose competitiveness at one point left two United players on their backsides. Others were not far behind in their desire to knock back the knockers.
Referee Steve Tanner threatened to undo the admirable effort by awarding the visitors a series of the most dubious free-kicks, all within striking range of Argyle's goal. Such was the perceived one-eyedness of the Somerset official that the sarcastic round of applause which greeted an eventual award to Argyle was one of the most ironic heard at Home Park for many years.
Argyle began the second half with, if not a bigger bang than the first half, then certainly a quicker one. Again Capaldi had a significant hand in the goal, jinking his way inside the penalty box from the left flank and knocking over a hard cross which Jagielka helped on to Norris at the far post.
His shot was blocked, but the rebound fell perfectly for Wotton and the home side's skipper pinged a rising shot past his opposite number from the edge of the area with such power and precision that Kenny probably would not have got near it either.
Argyle suddenly looked a different proposition. It would be pushing things to say they were playing with gay abandon, but they looked a world apart from the dispirited individuals of three days previously.
Nick Chadwick, who, with Evans, led the line like a veteran, came close to putting the issue beyond doubt twice in two minutes, with a close-range shot that was blocked by Nick Montgomery and then with an effort from distance which must have been a knee-trembler for Jagielka.
It is a football mantra that a 2-0 lead is the worst advantage to hold in football, but Argyle, who had not held such a margin since the last time they won – at home to QPR on Boxing Day and then for only four minutes – thrived on some rare breathing space.