An unstoppable 90th minute strike ended United's four-match unbeaten run.
Substitute Ryan Colclough won the game for Crewe after earlier goals from Lauri Dalla Valle and Jason Holt had left the scores locked at 1-1.
It was the fourth meeting between the sids this season and Crewe's first victory.
Howard returned in United's goal, replacing Turner, who dropped to the bench, it was boss Clough's only change following the drubbing of Scunthorpe United. Brayford retained the captain's armband against his former club.
Despite the midweek success, the crowd inside S2 was distinctly quiet early on, even with Flynn and Murphy prominent, collecting plenty of touches in the opening ten minutes, the latter was involved as Baxter's effort was blocked from a Davies lay-off.
Another Baxter effort was blocked after Brayford nodded back a Coutts corner just ahead of Crewe taking the lead with their first effort. Ikpeazu clipped through the defence and DALLA VALLE beat the offside trap and held off Brayford before lifting past Howard, into the roof of the net.
Too many loose passes prevented United immediately getting back into the game, whilst the narrow advantage ensured the visitors did not have to take risks. Brayford's stray pass on 21 minutes almost handed Crewe a second but from Grant's pass Ajose shot straight at Howard.
Referee Adcock intervened when United looked set to level on 23 minutes. From a short Davis back header, Davies challenged with Rachubka and as the loose ball dropped to Flynn to slot into the empty net, the whistle indicated a foul on the visiting goalkeeper. Soon after Brayford headed over from a Coutts corner as the Blades began to find some rhythm.
As half-time approached, United were attempting to find holes in the Crewe defence. Davies fired across goal and Coutts over from distance. However, the breakthrough was not to happen, with Baxter firing wide in injury-time, the last chance of the half.
Flynn was replaced by Ben Davies for the second half and namesake Steven was United's first yellow card, for a foul on Tate. Expectedly, the Blades kicked off on the offensive and the leveller arrived on 55 minutes. Turton could not cut out a Harris clip down the left and Murphy's pass inside saw HOLT beat his marker with his right foot before volleying home with his left, from an acute angle.
A neat inside pass from Baxter saw Murphy scuff the opportunity to put United ahead, whilst more stray deliveries saw momentum halted. A Davis shot for Crewe was comfortably saved by Howard.
Boss Clough made a double change with just over 15 minutes remaining, Doyle and Done taking over from Baxter and Steven Davies, and a couple of home chances followed. Murphy forced Rachubka into a near post save and from the resulting corner, Basham headed against the far post, despite being under pressure.
On 80 minutes a deflected Dalla Valle shot caused a moment of concern for Howard, whilst back on attacking duty, Davies almost attached himself to a powerful Freeman cross...and four minutes of added time was just being signalled when disaster struck.
Crewe won a corner from a free-kick and after a short routine between Ness and COLCLOUGH, the latter drilled a powerful effort through a crowd of bodies, which possibly left Howard unsighted. The goal left Bramall Lane stunned into the silence that the game started with.
There was one chance for United to level in added time but Coutts, after chesting down, hammered into the Kop, whilst Basham being added to the attack passed without incident.
Blades: Howard, Harris, Brayford, Basham, Flynn (B. Davies 45), Murphy, Baxter (Doyle 74), Coutts, Freeman, S. Davies (Done 74), Holt.
Unused: Alcock, McNulty, Turner, Adams.
Crewe: Rachubka, Davis, Turton, Atkinson (Stewart 76), Dalla Valle, Leigh, Ness, Tate, Ikpeazu (Colclough 71), Ajose (Inman 85), Grant.
Unused: Guthrie, Ray, Saunders, Waters.