Goals from Tom Huddlestone and Oumar Niasse either side of a Paul Pogba strike saw the Tigers pick up their first win over United since 1974, but it wasn't enough to overturn the 2-0 deficit from the first leg at Old Trafford as Jose Mourinho's side won 3-2 on aggregate.
United named a strong starting XI for the second leg, with the likes of David de Gea, Pogba and Zlatan Ibrahimovic all included, but that failed to prevent the visitors from putting in a lethargic first-half display.
Indeed, it was a slow start to the match from both sides until Harry Maguire had the first half-chance when he chested the ball down inside the area following a corner, only to balloon his resulting volley a long way over the crossbar.
De Gea was called into action for the first time two minutes later after Phil Jones had conceded a free kick in a dangerous area, but Sam Clucas's powerful strike was straight at the keeper, who punched the effort away.
United's threat at the other end was limited to a long-range Marcos Rojo strike which deflected narrowly wide as they went in search of a killer third goal in the tie, but it continued to be Hull who looked the most likely to open the scoring on the night.
The Tigers almost did exactly that in the 26th minute when Huddlestone's hooked ball into the box caught the United defence out and saw Michael Dawson collect it in the area, but Chris Smalling made a crucial last-ditch challenge before Niasse stung the palms of De Gea with a follow-up effort.
Hull kept the pressure on United with a Shaun Maloney free kick that flashed a few yards over the crossbar before finally breaking the deadlock with 10 minutes remaining in the first half.
The opener came from the spot after Jonathan Moss controversially adjudged Rojo to have tugged the shirt of Maguire from a corner, allowing Huddlestone to step up and bury his inch-perfect penalty into the bottom corner.
United almost responded immediately with their only real chance of the opening 45 minutes when Ibrahimovic was allowed to carry the ball a long way before looking to pass it into the bottom corner from 20 yards, but David Marshall got down well to turn it around the post.
The visitors did improve in the second half, but despite their increased pressure they struggled to create clear chances as Hull kept themselves in the tie.
A Smalling penalty claim was waved away by Moss, much to the chagrin of Mourinho, but the United manager's mood was eventually improved when his side levelled things up on the night with less than 25 minutes remaining.
The equaliser came from nothing as Huddlestone's attempts to track Marcus Rashford down the left channel only saw him divert the ball into a dangerous area, and Pogba reacted quickest to poke his finish into the bottom corner.
Hull's hopes of becoming just the second team in League Cup history to overturn a semi-final deficit of two goals or more suddenly seemed a long way away, and United almost sealed their place at Wembley once and for all with 17 minutes remaining when Rojo leapt highest inside the box to nod Jesse Lingard's corner against the crossbar.
Marco Silva's side came even closer to giving themselves a late lifeline from the resulting corner, but Niasse was denied his first goal in English football by the woodwork when his stooping header came back off the crossbar.
The former Everton striker would have better luck with five minutes remaining when he cut the aggregate deficit back to just one goal with a simple finish at the back post after David Meyler had turned Markovic's diagonal ball back across the six-yard box.
There was to be no fairytale comeback for the home side, though, as United held out to book their place in the final at Wembley on February 26, where they will meet Southampton.
Despite reaching the final, United see their 17-match unbeaten streak in all competitions come to an end, falling to their first defeat since November and their first loss at the hands of Hull in their last 14 meetings.
The Tigers, meanwhile, maintain their 100% home record under Silva having won three from three at the KCOM Stadium since his arrival.