The Spanish midfielder broke over Wolves keeper John Ruddy to settle the third-round replay once being place through on goal by Anthony Martial.
Wolves had AN early Pedro Neto strike dominated out by the video assistant referee for a handball within the build-up.
But it absolutely was a due win for the hosts, who had better possibilities during a tight game at Old Trafford.
Manchester United will next visit either Watford or Tranmere – whose third-round replay at Prenton Park on Tues was postponed due to serious rain.
Ole Gunnar Solskjaer’s side didn’t manage an effort on target in the goalless initial game between the 2 sides at Molineux on four Jan, however, they were rather more threatening during this match.
An excellent save by Ruddy prevented Mata from opening the scoring within the half, whereas Daniel James was additionally denied by the Wolves keeper.
After a high-tempo start, the game settled down – however Mata, 31, created the one moment of quality to seal triumph, rolling back the years with an inventive finish.