Rashford started on the bench in one of three United changes, came on as a second half substitute but was forced off after less than 15 minutes on the field. He is now a doubt for the trip to runaway leaders Liverpool in the Premier League on Sunday.
The brief cameo from United’s top goalscorer coincided with the only goal as Mata ensured Ole Gunnar Solskjaer’s men will travel to either Watford or Tranmere on January 26th.
Wolves put United out of the FA Cup last season and were unbeaten in five previous meetings between the sides since returning to the top flight in 2018.
Nuno Espirito Santo signalled his intent by making just one change from his side’s Premier League clash with Newcastle at the weekend as John Ruddy replaced Rui Patricio in goal and Wolves looked likely to inflict more pain on the Red Devils in the first half.
Pedro Neto’s strike was ruled out after a VAR review for handball against Raul Jimenez before Matt Doherty hit the post for the visitors.
At the other end, Ruddy made a fine save to deny Daniel James. The Welsh international fired just wide of the far post early in the second half.
Rashford had come off on Saturday after scoring twice in a 4-0 thrashing of Norwich, with Solskjaer admitting the England international is carrying “quite a few knocks.”
The Norwegian still ran the risk of introducing his star striker 25 minutes from time and within three minutes United led.
A rare moment of defensive disorganisation from Wolves allowed Anthony Martial to play Mata through on goal and the Spaniard coolly dinked the ball over the advancing Ruddy.
Rashford then went down holding his back and was replaced by Jesse Lingard.
His absence could be a huge blow to United’s fight for a top-four finish in the Premier League and in three cup competitions.
Cardiff set up a fourth-round visit to Reading by edging a seven-goal thriller against fourth-tier Carlisle United 4-3.