The Tigers were 2-1 behind to their Sky Bet League One opponents at half-time - the Blades having twice taken the lead through Jose Baxter and Stefan Scougall - but scored four times in the second half to book a return date at the national stadium against Arsenal on May 17.
Yannick Sagbo had briefly brought Hull level at 1-1 but the game was settled after the break with Matty Fryatt, Tom Huddlestone and former Blade Stephen Quinn scoring in an 18-minute burst.
Jamie Murphy set up a tense finale when he volleyed home in the 89th minute but David Meyler restored the two-goal cushion deep in added time.
Hull boss Steve Bruce, an FA Cup winner with Manchester United in his playing days, will take his share of credit for the turnaround, with Fryatt and Quinn scoring four and three minutes after being sent on as substitutes.
But the champagne moment of the afternoon was reserved for Huddlestone, who scored his side's wonderful third with a neat run and deft finish.
The game began at a brisk pace, although there was more nervous energy on show than composure on the ball.
Hull's early forays were unconvincing, Ahmed Elmohamady muscled out of a good position by Conor Coady and Huddlestone chancing his arm with a 40-yard chip that drifted harmlessly wide.
But the Blades were palpably up for the contest and were good value for their 19th-minute opener.
John Brayford, a dynamic presence pushing up from right-back, played provider with a devilish whipped cross from the right touchline and Baxter did just enough to turn it home from close range.
The striker nudged ahead of James Chester to make first contact and the ball squirmed home with Steve Harper unable to stop it.
Bootle-born Baxter celebrated by kissing his black armband, worn to mark the 25th anniversary of the Hillsborough disaster, and raising it to the sky.
Hull had plenty of time to cope with the setback but were deeply unimpressive for 20 minutes as they were outscrapped by United across the park.
Their equaliser, when it arrived in the 42nd minute, was as clinical as it was unexpected.
Jake Livermore exchanged a one-two with Elmohamady before finding Sagbo at the far post with a threaded pass across goal.
Having timed his run, the Ivorian made no mistake with his first-time finish.
On the touchline Bruce cut a relieved figure but his frustrations returned inside two minutes.
Murphy burrowed down the left flank for United and breezed past Liam Rosenior, who erred badly in allowing the winger to advance unchecked.
His cutback from the goal-line was begging to be buried and Scougall obliged with a fine finish.
Bruce hooked George Boyd and Maynor Figueroa at half-time, bolstering his attacking options with substitutes Fryatt and Sone Aluko.
It took Fryatt just four minutes to justify the gamble.
A Huddlestone corner fell for Chester and his mis-hit effort fell kindly for the substitute, who stroked home from six yards.
The Tigers could sense an opening and took the lead for the first time five minutes later.
It took a slice of class from Huddlestone, who exchanged passes with Meyler, shook off the attentions of Michael Doyle and shuffled the ball from right to left boot as he reached the penalty area.
By no means a regular scorer, he had the finish to match this time as he caressed the ball into the left corner.
Bruce went to the bench again in the 64th minute, Quinn replacing Sagbo, and the Irishman netted almost immediately.
Livermore claimed a second assist of the day with a dinked cross from the right and former Blade Quinn left Mark Howard no chance with a fine header.
At that stage there was a possibility that Hull could run riot, but that would have been unfair on Nigel Clough's side.
Indeed United invited yet more late drama as they poured forward and got their rewards in the 90th minute.
After pushing Hull deep in their own box the ball broke for Murphy, who volleyed home calmly.
The tantalising prospect of extra-time was briefly dangled but Meyler killed it with the eighth goal of a memorable day at HQ.