The Blades returned to winning ways in Sky Bet League One thanks to two early goals.
Scotsmen Ryan Flynn and Stefan Scougall - with his first for the Club - settled the contest with both goals inside the opening 20 minutes.
The victory also helped United leapfrog Shrewsbury in the table.
After the FA Cup heroics at Fulham, chief Clough made just one change with Baxter returning to the starting line-up and Porter dropping to the bench. Howard kept his place in goal following a clean sheet at Craven Cottage.
In the absence of the suspended Doyle, Collins again took care of the captain's armband but with the Blades kicking towards the Kop in the first half suggested that he had lost the pre-match toss.
Baxter, supported by the Scottish trio of Scougall, Flynn and Murphy, were the attacking options and it took just five minutes to create an opportunity. A neat move saw Baxter, Flynn and Brayford all involved before the latter's cross was headed clear by Mkandawire with Murphy lurking.
The lead, however, was not far away. On 11 minutes gone, McGinn found Scougall in space and his pass across the box saw Baxter step over, allowing FLYNN to come in from the right and drill home, despite a crowded area.
A second was just moments away and again it had a heavy Scottish influence. This time it was Collins who brought the ball out of defence and SCOUGALL took over, shooting into the bottom corner from 25 yards, although visiting goalkeeper Weale would not have been happy with his contribution.
The remainder of the half saw United clearly in the ascendancy and the more likely to be the next to score, although clear cut opportunities slowed. For the visitors, Parry's effort was well off target and then he provided a cross to the back post, which Mills almost attached himself to.
On 36 minutes, Flynn's powerful low cross saw Baxter control beautifully before shooting just wide and there was a spell of defending to complete ahead of applause from the crowd at the half-time whistle.
Early in the second half, following a 30-yard pass with the outside of Baxter's right boot, Murphy cut in from the left and fired over the angle of post and crossbar. A couple of Shrewsbury chances also kept them interested. Parry fed Taylor but he shot wide and Eaves blasted wide from distance.
The rain began to fall as the second half reached the midway point and Clough's first change was to replace McGinn with Miller. Professionally, Shrewsbury's second half possession was kept well away from Howard's goal, whilst the Blades dictated the pace without overly troubling Weale.
As the game entered the final ten minutes, the wind caused a problem for Howard but he managed to poke away ahead of Schmeltz, whilst at the other end Miller's shot was blocked by Mkandawire and Flynn shot over from the edge of the box.
Late on Coady was replaced by Porter, who slid the ball home on 88 minutes but was adjudged offside by the officials. That decision, however, could not take the shine of a deserved victory, which was confirmed by the final whistle soon after.
Blades: Howard, Harris, Brayford, Maguire, Flynn, Baxter, McGinn (Miller 68), Collins, Coady (Porter 83), Scougall, Murphy.
Unused: Hill, Paynter, Long, Kennedy, De Girolamo.
Shrewsbury: Weale, Jacobson (Schmeltz 48), McAllister, Fyvie (Grandison 79), Taylor, Mkandawire, Parry, Goldson, Woods, Eaves, Mills (Atajic 70).
Unused: Winfield, Bradshaw, Anyon, Hall.