The Blades are still without an away victory this season as Nigel Clough suffered his first defeat as Bramall Lane chief.
A blockbuster from Joe Jacobson and a second half goal on the break from Liam McAlinden secured the points for Shrewsbury.
However, the Blades were on the back foot from the moment Febian Brandy was shown a red card less than 15 minutes after coming on as a substitute - his second Blades sending off.
New loan capture White - Clough's first signing as Blades chief - started with McGinn dropping to the bench, where he was joined by Brandy, free from his virus. The favoured 4-4-2 formation continued but Lappin moved inside to central midfield to allow White to play in front of Hill on the left.
The conditions were going to be testing for both sides, it was quickly evident. The sun was out but the early drizzle made the surface difficult and the wind was quite strong, Shrewsbury having the advantage of the gust at their backs in the first half.
Shrewsbury almost took advantage of the conditions inside five minutes when Weale's clearance caught the breeze and sailed over the head of United's central defenders and as Long slipped as he looked to help out his team-mates, a header from Lopez drifted just wide of the open net.
A spell of Blades pressure gave hope, particularly after King's head down and Taylor's volley, which was blocked on 11 minutes. Lappin took into account the wind as he delivered a couple of corners and there were appeals for a penalty when he fell following a challenge from Woods but referee Attwell said no. White fired wide as he looked for a debut goal and Doyle and Taylor also saw further shots blocked.
As the game progressed through the half hour mark the Blades continued to be the more prominent force going forward. Doyle fired over after King had charged down a clearance from Woods and then Taylor was just wide with an acrobatic effort from King's cross.
On 39 minutes, though, the Blades were undone with a combination of brilliance and the conditions. The ball broke out to the left, 35 yards away from Long's goal but the Blades 'keeper was beaten by a first-time blast from JACOBSON, which flew into the top corner, undoubtedly helped by the wind.
There was still time for a riposte and it arrived in injury time at the end of the first half when Taylor slid the ball down the right channel for King but the striker fired over after entering the box unopposed.
Collins did not come out for the second half and Brandy's introduction prompted a reshuffle. Hill moved in to central defence, White to left back and Brandy slotted in at left wing.
McMahon tried to use the wind to United's advantage with an early second half shot but Weale was well-placed and then a low White delivery from the left flew across the box.
However, just as the Blades were readying Miller for action, plans needed to change. Just before the hour Brandy went down and was denied a free-kick but the aftermath was significant. Goldson deemed it appropriate to take up the matter with the Blades substitute and after pushing him away, Brandy was shown a red card by referee Attwell.
A numerical advantage gave Shrewsbury the opportunity to come out of defence and Hill denied Bradshaw, whilst McMahon and Maguire made significant blocks.
Miller was introduced for King and the Blades continued with two up front, although the substitute was booked for a late challenge less than ten minutes after being introduced.
As the final ten minutes approached, McGinn, a former Shrewsbury loan player, was introduced but committing players forward ultimately ended in problems. The home side broke away and in a three-on-on situation, McMahon the only defender, Bradshaw laid off to McALINDEN to fire into the roof of the net.
The Blades continued to battle and Weale was forced to make a save from Taylor's shot, whilst Miller was unable to connect with a Lappin cross.
Four minutes of added time just served to prolong the agony for the Blades and there were no further incidents of note before the referee's final whistle.
Shrewsbury: Weale, Jacobson (Goldson 45), Winfield, Bradshaw (Summerfield 87), Wroe, Taylor, Wildig, Mkandawire, Woods, Reach, Lopez (McAlinden 73).
Unused: Jones, McAllister, Parry, Anyon.
Blades: Long, White, Maguire, Hill, Flynn (McGinn 78), Doyle, Taylor, Collins (Brandy 45), King (Brandy 63), Lappin, McMahon.
Unused: Howard, Baxter, Cuvelier, Coady.