Mata was played clean through by Anthony Martial and calmly chipped goalkeeper John Ruddy for the only goal against Wolves who knocked United out of last season's FA Cup and had a 10th-minute effort ruled out by VAR.
The first match at Molineux between the teams, both in the last 32 of the Europa League and with plenty of tired legs, had ended goalless with United failing to land a single shot on target.
United's celebrations were muted, however, by an injury to their top scorer Marcus Rashford who came on as a second-half substitute for Daniel James and played only 16 minutes before limping off injured.
"That backfired, but you've got to go for them once in a while," manager Ole Gunnar Solskjaer told BT Sport.
The injury left the England striker a doubt for Sunday's Premier League clash against runaway leaders Liverpool at Anfield.
"It seems like he's got a knock there and he couldn't run, so let's test him in the next couple of days. Hopefully, he'll be OK for Sunday, I don't know," said Solskjaer.
In the night's other replay, Cardiff City beat fourth tier strugglers Carlisle United 4-3 in a livelier game to set up an all-Championship (second tier) clash with Reading in the fourth round.
United will next play either Premier League rivals Watford, last year's runners-up, or third tier Tranmere Rovers whose replay on Tuesday was postponed due to a waterlogged pitch at Prenton Park.
Solskjaer's side were let off the hook after 10 minutes when a VAR review showed the ball had bounced off Raul Jimenez's hand before Portuguese teenager Pedro Neto scored for Wolves.
The visitors also went close from a corner when Matt Doherty flicked his header onto the post.
Even if the score remained goalless at the break, there had still been five shots on target with United providing three of them.
"I think you can see there's two teams there that towards the end were tired. End-to-end, an open game which sometimes doesn't suit us. We wanted to control the game more," said Solskjaer.
"Juan Mata is different class. He's got the skill, the composure and even pace to get through there."
Wolves manager Nuno Espirito Santo said his side had not defended the goal well but it had been a good game and he was frustrated with the outcome.
"That's VAR, you cannot do anything. We are celebrating non goals. It doesn't make sense," he added.