The Worthington Cup semi-finalists had to survive early Preston pressure before the breakthrough came in the 58th minute.
Former Sheffield Wednesday striker Richard Cresswell saw his shot smothered on the line by Paddy Kenny before centre back Robert Page headed a dangerous Eddie Lewis cross away as the Lancashire side looked for an early goal.
Tyrone Mears almost got the breakthrough in the 17th minute. Page failed to halt his searing run as the Preston midfielder fired in a low strike from eight yards.
Kenny managed to block the shot and when the ball was pushed across the face of the goal, Cresswell was foiled by the United keeper who scooped his close-range effort off the line.
David Healy had a long-range strike cut out by Kenny before United broke quickly - Peter Ndlovu firing his 30-yard effort straight at the waiting David Lucas with Steve Kabba in space.
Former Crystal Palace frontman Kabba almost snatched the lead in the 31st minute as he strained to meet Jon Harley's cross, but Lucas finger-tipped his header over the bar.
Dangerman Healy fired a right-foot shot wide of the target from Alexander's free-kick and was inches away from connecting with a Mears cross minutes after the restart.
United swept ahead in bizarre fashion in the 58th minute after an astonishing decision by referee Frazer Stretton,Cresswell was clearly bundled off the ball by Shaun Murphy in the United box but no penalty was given.
The Blades swept upfield, Michael Tonge firing in a vicious cross from the right and Alexander, trying to clear, produced a diving header to slam the ball into his own net.
The incensed Preston team surrounded Mr Stretton, protesting over the penalty decision. Colin Murdock was booked for dissent and Rankine was shown a red card for remarks made to the referee.