In the team's first semifinal in the competition since 1930, Steve Bruce's men recovered from a timid start to score three goals in 18 second-half minutes and book a final clash with Arsenal on May 17.
Jose Baxter, an unused substitute for Everton in the 2009 final, put United in front 19 minutes in, getting on the end of John Brayford's right-wing cross to poke the ball home from six yards, although the final touch appeared to come off Hull defender James Chester.
Yannick Sagbo - selected as Hull's lone striker in the absence of the cup-tied Nikica Jelavic and Shane Long - equalized shortly before the break, sliding in to convert Jake Livermore's beautifully weighted pass.
Nigel Clough's men responded immediately, taking a 2-1 lead into halftime when Stefan Scougall converted Jamie Murphy's cutback to set the underdog up well for the second half.
Hull was dominant after the interval, though, with former Sheffield Wednesday loanee and halftime substitute Matty Fryatt leveling the score by sweeping home from close range just after the break.
Tom Huddlestone finished off a fine move with a composed finish to put the Premier League side in front and Stephen Quinn - another inspired Bruce substitution - headed a fourth to put Hull two up.
Jamie Murphy pulled one back late on to make things interesting for a moment, but David Meyler put the result beyond doubt deep into stoppage time to end the underdogs' remarkable run through the competition.
The win sets Hull up with a showdown against Arsenal in next month's final.
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