Their fans are split over whether their name should feature City or Tigers but one thing Hull supporters can now agree on is that it is wonderful to be able to call their team FA Cup finalists. Never before in the club's 110-year existence have they had that pleasure.

For a while here it looked like their wait would go on. Deservedly trailing 2-1 at half-time to a Sheffield United side intent on once again belying their League One status, Hull re-emerged for the second half as a far more vibrant team and struck three quick goals before valiant United threatened a fightback of their own. Not until David Meyler's goal in stoppage time were the Premier League side guaranteed victory in this thrilling semi-final and a return to Wembley, where they will assume the more familiar role of underdogs in the final against Arsenal.

United were left to reflect on a superb Cup run that included wins over four teams from higher leagues and an excellent first half here. They came very close to becoming the first third-tier team in over a century to reach the FA Cup final.

The Blades started far sharper than their opponents, who seemed petrified by the occasion and their status as heavy favourites. Despite being more brawny, Hull were being bullied by a team playing with no fear. They were also being dominated tactically. Nigel Clough has inspired a revival in United's fortune since taking charge in October and here his design helped give his side the advantage. Playing without a specialist striker – the midfielder Jose Baxter instead served as a false No9 – United swarmed all over Hull. With Yannick Sagbo struggling to make any impact up front in the absence of the Cup-tied Shane Long and Nikica Jelavic, and Hull utterly devoid of width, the top-flight team were bereft of ideas. And they were getting even the basics all wrong.

United had not forced Steve Harper into a save but still deserved the lead when it came in the 19th minute. Hull negligence allowed John Brayford to receive the ball back after taking a throw-in and curl a cross in from the right. Baxter beat James Chester to it and nudged the ball into the net from close range.

Three minutes before half-time Hull produced the first vignette of Premier League class and drew level. Jake Livermore swapped a one-two with Ahmed Elmohamady before delivering a low ball into the box. Sagbo peeled off his marker and finished into the roof of the net.

Relief swept through Hull – but they reclined too much. United regained the lead within two minutes. Murphy showed greater strength as he surged past Liam Rosenior and crossed for Stefan Scougall, who darted between two defenders to fire into the net. "We were awful in the first half," admitted Hull's manager, Steve Bruce. "We couldn't pass, couldn't defend, couldn't do anything. We had to change it at half-time."

They also had to find rousing words, a duty assumed during the break by the captain, Curtis Davies. "It's a long time since I've been in a dressing room where the captain takes over," said Bruce. "I just had a cup of tea and let him get on with it. There were a few tactical changes but the bollockings came from him."

The tactical changes entailed introducing two strikers and they paid off quickly. One of the new arrivals, Matty Fryatt, equalised almost immediately, sweeping the ball into the net from close range after a Tom Huddlestone corner had provoked a scramble in the United box.

Huddlestone, bypassed in the first period, then put Hull in front. After collecting the ball in midfield he fed Meyler before lumbering forward to receive the return pass, shrug off Michael Doyle and finish from 15 yards.

Hull's fourth goal was scored by Stephen Quinn, a former United trainee who had been applauded before the match by the fans in red and white for the seven years he spent at Bramall Lane. Quinn, Bruce's third substitute of the game, headed into the net in the 67th minute after nifty work down the left by Meyler and Livermore.

United, defiant to the end, refused to accept their fate without a fight and made Hull sweat by cutting the deficit in the 90th minute, Jamie Murphy rifling the ball into the net from 12 yards after a strong run down the right by Bob Harris. But Meyler made sure of Hull's victory by crowning a counterattack with an emphatic finish in stoppage time. "I'm proud of the team and that feeling will grow over the coming weeks as we look back at what we've achieved," said Clough.

Bruce also praised United and his own team's comeback, before turning his thoughts to the final. "All the pressure will be on Arsenal because they haven't won anything in eight years," he said before admonishing himself: "I'm starting all the bullshit now: how sad is that?"

Man of the match Jake Livermore (Hull)