Mata and Fellaini seal the win
All eyes were on United captain Wayne Rooney, who is chasing a record 250th goal for the club, but it was Mata who broke the deadlock in the 56th minute before substitute Fellaini added a late second. It puts Jose Mourinho's men in the driving seat ahead of the return leg on January 26, although it was not until Fellaini headed home in the 87th minute that they finally put the game beyond Hull.
Nevertheless, they have now won nine games in succession and look in fine fettle ahead of arch rivals Liverpool's visit in the Premier League on Sunday.
Hull's new manager Marco Silva, taking charge of his second game, was able to name only six substitutes and lost both Markus Henriksen and Josh Tymon to injury. Despite Mourinho making seven changes to the team that had outclassed Reading 4-0 in the FA Cup on Saturday, United picked up where they had left off, peppering the Hull goal with shots.
Hull goalkeeper Eldin Jakupovic had to produce a one-handed save to thwart Mata inside two minutes and moments later Andy Robertson stopped Rooney applying the finishing touch to Marcus Rashford's shot. The presence of David Meyler and Tom Huddlestone in defence and the fact there were only six players on the Hull bench told the story of the visitors' injury problems and there was more to come.
Injury hell for Hull manager
Henriksen was left clutching his right shoulder in agony after being knocked over by Paul Pogba and had to go off in the 16th minute, with Abel Hernandez taking his place. United, missing the ill Zlatan Ibrahimovic, continued to swarm forward, Mkhitaryan twice shooting wide, Rashford lashing over and Jakupovic shovelling a long-range Pogba effort over the crossbar.
But Hull, the Premier League's bottom club, began to offer an attacking threat of their own as half-time approached in the club's first ever League Cup semi-final match. Adama Diomande gave United a scare when he headed against the post, only for an offside flag to then go up, and both Snodgrass and Hernandez tested David de Gea before the interval.
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Rooney's moment of history looked to have arrived in the 51st minute when Pogba's fine, flighted pass set him racing in on goal from the right, but he drilled his shot inches wide of the far post. Instead it was Mata who provided the breakthrough, stabbing in on the volley at the back post after Antonio Valencia's cross from the right was headed back across goal by Mkhitaryan.
After Diomande had volleyed over acrobatically for Hull, Pogba came within a whisker of doubling United's lead, his curling free-kick clanging against the left-hand post and bouncing away. It seemed Hull would escape with only a one-goal deficit, but with three minutes remaining Fellaini met Matteo Darmian's cross with a looping header that Huddlestone could only help over the line. Silva's injury woes mounted yet further in stoppage time when 17-year-old left-back Tymon had to limp off.
Liverpool, last season's beaten finalists, visit Southampton in the first leg of the second semi-final on Wednesday.