Hull survived a number of dangerous first-half moments to keep the score level at the interval, but Mata and Fellaini registered in the second period to hand Man United, who have now won their last nine games in all competitions, a two-goal advantage ahead of the second leg at the KCOM Stadium on January 26.
Man United boss Jose Mourinho made seven changes to the team that started Saturday's 4-0 win over Reading in the third round of the FA Cup, but club captain Wayne Rooney kept his spot in the XI as he looked for the one goal that would make him the club's all-time leading scorer ahead of Sir Bobby Charlton.
Zlatan Ibrahimovic missed out altogether through illness, which meant that there was a start at centre-forward for Marcus Rashford, while Paul Pogba and Henrikh Mkhitaryan were among those to return to the team.
As for Hull, first-choice centre-backs Curtis Davies and Michael Dawson both missed out through injury, while Jake Livermore was unavailable due to a virus, leaving the Tigers short of options in a defensive area.
David Meyler, Harry Maguire and Tom Huddlestone joined Andrew Robertson at the back, while Ryan Mason and Markus Henriksen kept their spots in midfield as the visitors went in search of a shock first-leg win.
Man United had a super chance to take the lead inside the first two minutes when Matteo Darmian's clever cross found Mata on the edge of the Hull box, but the Spaniard was denied by a smart save from Eldin Jakupovic.
Darmian set up another chance for the Red Devils in the seventh minute when his low cross found Rashford, who swiped the ball towards Rooney, but Robertson just did enough to make the block. Hull's first shot on target then arrived in the 10th minute when Maguire bullied his way through a host of challenges before having a pop, but David de Gea collected.
Mkhitaryan should have sent Man United ahead in the 19th minute when brilliant work from Mata provided the Armenian with a shooting chance inside the Hull box, but the summer signing placed the ball wide of the near post, before firing over the crossbar from a more central position five minutes later.
Pogba brought a smart save from Jakupovic in the 28th minute as the home side came close once again, before Rashford fired just over the crossbar as it somehow remained goalless approaching the final third of the first period.
Hull actually had a brilliant opportunity of their own in the 31st minute when Adama Diomande found himself free following a deep set piece into the Man United box, but the forward's header hit the post as the Red Devils survived a really dangerous moment.
Abel Hernandez and Pogba both had decent attempts at either end in the latter stages of the first period, but the two teams headed down the tunnel on level terms as the home side failed to make the most of a number of clear-cut opportunities and more than 70% possession.
A sensational pass from Pogba released Rooney in the 51st minute and the striker's low effort looked to be picking out the far corner for the scoring record, but the ball just flew wide of the far post as Hull continued to survive.
Man United refused to stop creating opportunities, however, and the Red Devils finally made the breakthrough in the 56th minute when Mata tapped home from close range after Mkhitaryan had headed an Antonio Valencia cross back across goal.
There was to be no record-breaking goal for Rooney on the night as Anthony Martial replaced the club captain before the hour, fresh from his impressive performance against Reading in the FA Cup at the weekend.
Diomande tried the spectacular 25 minutes from time after meeting a deep cross from Meyer, but the Hull attacker's effort missed the crossbar during what was a decent period for the visitors after falling behind.
Jesse Lingard replaced Mkhitaryan in Man United's second change of the match in the 71st minute, and two minutes later Pogba hit the post with a free kick as the woodwork came into action for the second time on the night.
Shaun Maloney came off the bench for Hull 15 minutes from time and had a shooting chance less than 30 seconds after being introduced, but the experienced campaigner missed the target from outside the Man United box.
Rashford's pace proved key in the latter stages as Hull's legs continued to tire, but the visitors were still a threat and Valencia had to be alert to head a Robert Snodgrass cross behind under pressure from Sam Clucas.
Man United scored their second of the night in the 87th minute, however, when Darmian delivered a looping cross towards the head of Fellaini and his effort dropped over Jakupovic as the Red Devils ensured that they would take a 2-0 lead into the second leg on January 26.
Next up for Man United is a home Premier League match against Liverpool on Sunday afternoon, while struggling Hull will look to move off the foot of the table when they welcome Bournemouth one day earlier.