A tremendous hard-working performance laid the foundations for a fantastic victory at The Hawthorns.
Kieran Dowell's first half header was the difference but Dean Henderson made a fantastic save in the final minute of added time to preserve the three points.
Blades chief Wilder made three changes for the early evening fixture; all three at the top of the pitch where Dowell, McGoldrick and Sharp returned. Duffy, Madine and Hogan dropped to the bench and were joined by Basham, eligible again after suspension.
The first chance, on five minutes, saw Rodriguez head over a Holgate cross from the right for the home side. United's players were more clinical with the first serious opportunity for the visitors. On the break, Norwood saw space on the right and involved Freeman and the overlapping Cranie, who delivered the cross for DOWELL to head home his first Blades goal - goalkeeper Johnstone well beaten.
Johnstone, though, denied United a second on 19 minutes. Great build-up play from Dowell resulted in him laying off for Johnson, whose right-foot curler was heading for the bottom corner before the goalkeeper stretched to turn the ball around a post.
Holgate cut inside both Johnson and Dowell but his curler drifted wide of Henderson's far post, with the aid of a deflection, whilst Johnstone was called upon once again to deny McGoldrick at the near post.
Albion tried to build a bit of pressure but Egan prevented Phillips from threading a ball through to give the home side a clear sight on goal and then Gayle, Dawson and Harper all had efforts.
The final effort of the half, though, was United's but Johnstone nicked the ball off Egan's head from Norwood's corner. Egan didn't reappear for the second half and Stearman took over in the heart of United's defence.
United's committed approach was evident in two incidents at the start of the second half. Great covering from Johnson prevented West Brom getting in following a Blades free-kick and then Stevens blocked a Gayle volley.
Madine and then Basham were introduced as United, unsurprisingly, tired as the work-rate continued into the closing moments. Albion got around the back of the defence on 72 minutes but Henderson was well-placed to collect Gayle's low cross. Gayle also failed get a substantial touch to a Gibbs cross as the home side continued to try and apply the pressure.
United remained a threat and Sharp's header was wide following a patient move on the right, whilst Madine fired straight at Johnstone after Dowell's break.
On 84 minutes West Brom had the ball in the net but Rodriguez clearly directed the ball home with his hand and was subsequently booked and a minute later Henderson was quickly off his line to snuff out Gayle, who appeared to be offside.
Five minutes of added time brought anxious times for the 2,673 Blades fans inside the Hawthorns and in the final moment Henderson saved the best for last, displaying perfect positioning to deny Gayle, who was in the clear after the ball skimmed off the head of Stevens.
Referee Langford's final whistle was a welcome sound as the Blades finished the weekend's round of matches in second position.
Albion: Johnstone, Gibbs, Livermore (Edwards 87), Phillips (Montero 64), Gayle, Barry, Rodriguez, Dawson, Hegazi (Adarabioyo 65), Harper, Holgate.
Unused: Bartley, Johansen, Bond, Field.
Blades: Henderson, Stevens, Fleck, Dowell, Sharp, Egan (Stearman 45), Norwood, McGoldrick (Madine 65), Freeman, Johnson, Cranie (Basham 74).
Unused: Hogan, Coutts, Duffy, Moore.