The League One side took the lead when John Brayford delivered a superb ball into the area which landed perfectly into the path of Jose Baxter, who tapped the ball past keeper Steve Harper.
The scores were levelled just before half-time when Yannick Sagbo emphatically finished off a well-worked move, but United restored the lead moments later when Stefan Scougall fired high into Harper's net following a powerful run from winger Jamie Murphy.
City boss Steve Bruce brought on Sone Aluko and Matty Fryatt at half-time, and the latter repaid the faith shown in him by stroking in an equaliser on the 48th minute. Five minutes later, Tom Huddlestone broke into the Blades' box and curled home a superb third for the Tigers - sending them in front for the first time.
Former United winger Stephen Quinn headed in a fourth on 66 minutes and, despite Jamie Murphy's late goal for the Blades, David Meyler netted a fifth which confirmed Hull's place in the final against Arsenal on May 17.
The game began at a brisk pace, although there was more nervous energy on show than composure on the ball.
But the Blades were palpably up for the contest and were good value for their 19th-minute opener.
Brayford, a dynamic presence pushing up from right-back, played provider with a devilishly-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 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 Ahmed Elmohamady before finding Sagbo at the far post with a threaded pass across goal and the Ivorian made no mistake with his first-time finish.
Murphy burrowed down the left flank for United and breezed past Liam Rosenior, and 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 Aluko.
A Huddlestone corner fell for Chester and his mis-hit effort fell kindly for Fryatt, who stroked home from six yards.
It took a slice of class from Huddlestone to score the third, when he 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.