The hosts had spurned several chances to put the tie to bed, with Paul Pogba rattling the post, before substitute Fellaini gave Jose Mourinho's side some breathing space when he rose at the far post to head home with three minutes remaining.
The victory kept the Portuguese coach on course for a first trophy at the club and continued United's superb recent form with a ninth successive victory in all competitions, but the demanding Mourinho was far from satisfied with what he saw.
"I was a bit disappointed yes," he told Sky Sports.
"I think we were a bit sloppy, overcomplicating things, always delaying the decision, giving them time to regroup... In the second half we all improved a little bit, just a little bit."
INJURY CRISIS
In only his second game as Hull boss, fellow Portuguese Marco Silva had to deal with an injury crisis that left him with only 14 fit senior players and there was a makeshift look about them as United took control early on.
Mata drew an excellent save from Hull keeper Eldin Jakupovic after two minutes, while Henrikh Mkhitaryan sidefooted wide after being set up perfectly by the Spaniard.
The visitors, whose injury woes got worse when Markus Henriksen went off after 19 minutes with a shoulder problem, did have arguably the best chance of the first half, however, when Adama Diomande headed against the base of the post.
Wayne Rooney, seeking to become United's outright record scorer, crashed a shot just wide five minutes after the break and was later substituted, but the deadlock was soon broken.
Antonio Valencia's deep cross to the far post was headed back across goal by Mkhitaryan and Mata ghosted in to volley home from close range.
United almost doubled their lead as Paul Pogba rattled the post with a free kick, but the frustration levels were rising until Belgian Fellaini came off the bench to make his mark and provide a cushion for the second leg on Jan. 26.
Southampton and Liverpool contest the other semi-final, with the first leg at St Mary's on Wednesday.
(Reporting by Toby Davis; Editing by Ken Ferris)
