In their first semi-final 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 but Yannick Sagbo - selected as Hull's lone striker in the absence of the cup-tied Nikica Jelavic and Shane Long - equalised shortly before the break.
Nigel Clough's men responded immediately, taking a 2-1 lead into half-time when Stefan Scougall converted Jamie Murphy's cutback.
Hull were dominant after the interval, though, former Sheffield Wednesday loanee and half-time substitute Matty Fryatt levelling the scores once more before Tom Huddlestone finished off a fine move with a composed finish.
Former United player Stephen Quinn - another inspired Bruce substitution - headed a fourth, and while Murphy pulled one back, David Meyler put the result beyond doubt deep into stoppage time to end the underdogs' remarkable run.
After a cagey opening United took the lead in the 19th minute, Baxter 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.
Baxter, who was born in Liverpool, wheeled away in celebration, kissing his black armband commemorating the Hillsborough disaster.
Scougall fired high and wide from the edge of the area five minutes later as the League One side continued to have the better of the play against their top-flight opponents.
But Hull drew level with their first meaningful attack of the game three minutes before the break, Sagbo sliding in to convert Jake Livermore's beautifully weighted pass.
Parity lasted just two minutes, however, as Scougall thumped home from eight yards after an excellent run down the left from Murphy.
Bruce brought on Fryatt and Sone Aluko at the break and the change bore fruit four minutes after the interval.
United failed to clear a corner and Chester stabbed the ball goalwards, where Fryatt was well placed to sweep into the net from close range.
Huddlestone put Hull in front for the first time five minutes later, exchanging passes with Meyler before evading the challenges of Neill Collins and Michael Doyle to plant a finish beyond Mark Howard.
Quinn headed in from eight yards midway through the half after Meyler and Livermore combined down the right-hand side, before Murphy reduced the deficit with a close-range finish.
Hopes of a United comeback were dashed, though, when Meyler broke free to make it two wins out of two for Hull at the new Wembley - having prevailed there in the 2008 play-off final - and secure their berth in next month's final at the national stadium.
