Micky Adams left Sheffield with his head held high, but a goal from Tom Pope proved to be not enough as the home side snatched a victory with two goals in the dying minutes of the match.
Vale were boosted by an estimated 3,000 travelling supporters who had made the visit to South Yorkshire for the FA Cup second round tie.
Clayton McDonald was passed fit to start while captain Doug Loft and new signing Calvin Andrew made the bench. Kingsley James lined up against his former club.
Neal had to save well from McDonald deflection in the opening stages, but Vale rallied with two chances for leading scorer Tom Pope in the first twenty minutes- one going wide and one forcing a save from home keeper Long.
But it was largely an even affair until the game burst into life after the half-hour mark.
Pope first set up strike partner Ben Williamson who broke free and forced a good save from Long before Pope himself gave Vale the lead. Sloppy play in the United defence gifted the ball to Vale and Pope made no mistake with a low drive into the right-hand corner of Long’s net.
The home side appealed for a penalty in vain as the break approached but the appeals were turned down as Vale took a half-time lead.
The home side rallied after the break but Neal pulled off a great save in the opening mins. Then Miller spun and shot narrowly wide of the post as the Blades pressed. And Vale were grateful for a crashing challenge from Duffy as a United player threatened to break clear.
Nevertheless, Vale were looking capable and dangerous on the break with Pope linking things up well and the pace of Vincent and ever-willing Williamson threatening.
But the home side had the next good chance with Miller mis-kicking as he found space in the Vale box. And Neal was quickly off his line to deny Blackman moments later.
Vale were hanging on at times with the defence clearing their lines as the home side pushed for an equaliser.
Vale then made a double change that saw Louis Dodds and Calvin Andrew, making his Vale debut, come on.
United had a glorious opportunity to equalise with ten minutes left, but the unmarked Miller’s header bounced up and over the bar. Then the dangerous Blackman hit the side netting after a mazy run.
Vale then had claims for a penalty as Pope’s shot was blocked by a United arm inside the box, but the referee failed to give it.
And in the last minute of normal time, Vale were denied a notable victory Â when Miller’s header gave Neal no chance.
And in a dramatic twist, as the game reached the final minute of added time, Miller pounced on a rebound after Neal had parried a shot to put United in front.
It was rough justice on Vale, who at least can be proud of a good first-half display against opponents from a higher division.
Vale: Neal, Duffy, McDonald, McCombe, Yates, Myrie-Williams (Dodds), Burge, James, Vincent (Taylor), Pope, Williamson (Andrew)
Subs: Johnson, Loft, Davis, Shuker