This fixture lived up to its Yorkshire derby billing and honours were even after 90 minutes of hard graft.
The Blades raced into a 2-0 half-time lead with goals from Jamie Murphy and Harry Maguire but Gary Jones and James Hanson levelled after the break.
Despite losing the lead, Blades fans applauded at the final whistle after an exhilarating, whole-hearted performance.
Loanee Paynter was handed a home debut from the start as chief Clough made two changes from the previous week's defeat. Murphy was recalled and Porter and Coady dropped to the bench.
Bradford's squad included three former Blades - Reid, Gray and Yeates.
Within three minutes both sides had swung in a free-kick from the left and it was Bradford who went closest, Hanson meeting the delivery from Jones but Long was ideally placed.
The visitors started well but were stung with the game's first piece of quality, which arrived on eight minutes. An intricate passing move involving four Blades enabled McMahon the space he needed to drill a delivery low across the box to where MURPHY stole a yard on his marker to score.
Bradford's response was instant and they went close a number of times within the next five minutes. Firstly, McLean's effort was deflected wide by Maguire and then the same striker headed over the resulting corner.
Soon after, from another Jones corner, Hanson's powerful header struck the crossbar and the rebound, from McHugh, was cleared off the line by Baxter.
The opening 20 minutes was all-action. Paynter had an effort blocked after Maguire and Collins challenged in the air and Baxter tried to get on the end of a McMahon cross after another neat passage of play from the Blades.
To show the home side were also a threat at set-pieces, Maguire's header from Baxter's corner needed touching over the crossbar by McLaughlin, whilst around the half-hour mark there was a lengthy stoppage as Reid received treatment and was subsequently stretchered off, all four corners of Bramall Lane applauding as he left the field. One former Blade took over from another as Yeates was the replacement.
United's attacking prowess from corners was proved as the referee's watch ticked around to 40 minutes. MAGUIRE again met Baxter's corner and this time a combination of goalkeeper McLaughlin and Jones, positioned next to a post, couldn't keep the ball out.
The remainder of the half, including the four added minutes, saw the Blades on top. Baxter headed wide after Flynn linked up with Paynter and then McLaughlin scrambled across his line to keep out another Baxter shot. Also before the break, Murphy was crowded out in a scramble on the line and a Flynn shot was blocked.
Like the first half, the second period began at a pace. Hanson and McLean both failed to convert a Yeates cross and Baxter turned a centre from McMahon over. A bit of fortune then got the visitors back in the game, on 56 minutes. Hanson and McLean were both involved before a shot from JONES took a major deflection, wrong-footing Jones and spinning into the net.
An effort from Yeates moments later also took a deflection, this time to Long's benefit, but Bradford were soon level. Just after the hour, McGinn was penalised for a foul on McLean and from a Jones free-kick, Long diverted the ball at HANSON, the ball dropping in the net.
Bradford's second was the signal for the Blades to get back on top. With Flynn and Murphy both causing problems in wide areas, Porter replaced a tiring Paynter and now it was the visiting side's time to be pressured.
Flynn found McGinn, whose shot was turned around the near post by McLaughlin and from the resulting corner, Maguire's flicked header was wide. Murphy beat Darby twice but saw his shot blocked, whilst further efforts from Flynn and McGinn saw last-ditch defending keep the ball out.
Miller replaced an injured Baxter and a McMahon free-kick was wide. Late on there were penalty shouts at both end and Coady took over from McGinn. A Miller run, drawing the goalkeeper out of the goal, resulted in Murphy firing over from a difficult position.
Five added minutes gave both teams hope of snatching a winner but after an entertaining fixture it was possibly the only five minute period where there wasn't a chance!
Blades: Long, Maguire, Hill, Flynn, Doyle, Baxter (Miller 79), McGinn, Collins, Paynter (Porter 61), Murphy, McMahon.
Unused: Howard, Coady, McFadzean, Whitehouse, Kennedy.
Bradford: McLaughlin, Darby, Reid (Yeates 31), Kennedy (Doyle 72), Hanson, Thompson, McHugh, Jones, McLean (Gray 90), Bates, McArdle.
Unused: Oliver, Jameson, Taylor, Graham.