A strike just before the hour by Jason Holt looked to have given the Blades a victory that would have meant a third tilt at the promotion deciders in four years had been sealed.
However, a tremendous finish from Clarke six minutes from the end ensured honours ended even.
Bradford played five minutes of stoppage time with ten men following Rory McArdle’s dismissal but United rarely looked like grabbing a late winner.
Nigel Clough’s men are all but there in terms of a top six place. Mathematically, however, Rochdale can still make up the nine points advantage United possess due to having three games in hand.
Bradford, clearly smarting from the midweek 6-0 thumping by Bristol City, started strongly and twice went close inside the first three minutes.
Andy Halliday was the first to shoot an inch or so wide, closely followed by Billy Knott after neat inter-play from Billy Clarke and James Hanson.
Clarke also came the closest to breaking the deadlock for the Bantams in the first half with a 16th minute shot that Chris Basham did well to clear off the line with goalkeeper Mark Howard well beaten.
United also had several moments of promise in an open first half as Jamie Murphy shot over early on and Paul Coutts saw a drilled effort blocked by Rory McArdle.
Steven Davies fired a free-kick straight into the defensive wall just before the break but the home side’s persistence paid off 12 minutes into the second half.
Neat play from Jamie Murphy on the right created the opening and, after Michael Doyle had a shot blocked, Holt swept in from close range.
City’s impressive response saw plenty of questions asked of the home defence as, first, James Meredith was denied a shooting opportunity and then Chris Basham cleared a Mark Yeates cross just as Hanson looked ready to pounce.
Yeates also had a volley that Howard did well to hold before then curling an 80th minute effort that Brayford headed clear of danger.
It looked like being United’s day but then, just six minutes from the end, Clarke fired in from the edge of the area to give the visitors a deserved point.
McArdle’s dismissal - for a foul on Steven Davies, his second of the game - made it an anxious finale but, in truth, the Blades rarely looked like snatching a winner.