Wolves dominated possession in the first half without scoring, but it was a 67th-minute United counter-attack that allowed Juan Mata to strike the only goal of the game.
Ole Gunnar Solskjaer drafted in goalkeeper Sergio Romero, with forwards Mason Greenwood and Daniel James coming into his starting XI. In-form striker Marcus Rashford was on the bench.
Stopper John Ruddy was handed a start by Nuno Espirito Santo, and the deputy 'keeper quickly denied James in the opening moments.
Wolves believed they had grabbed the lead after nine minutes when Pedro Neto fired home after a mistake by Fred, but VAR disallowed the goal for handball against Raul Jimenez. The ball deflected off the Mexican's arm in a challenge with Harry Maguire before the goal-bound strike.
Fred almost atoned for his earlier error with a strike that missed the target, but United continued to struggle to find their rhythm in the first half. Wolves looked dangerous on the front foot. The hosts were slow to respond to the visitors' quick-thinking, and Romero was the busier 'keeper.
United appeared rigid in midfield, but the counter-attack threat was present for the Red Devils. Mata spurned a good chance to take the lead midway through the first half, but Ruddy was equal to his strike.
A set piece undid United after 29 minutes, as Jimenez's header beat Romero but hit the post.
United's technical shortcomings from previous games were apparent again. The away side exploited their opponents when they lost the ball and appeared the more likely to score.
Solskjaer's men came out of the blocks quicker in the second half, and James and Mata both had good chances of breaking the deadlock.
However, Anthony Martial was more isolated without the support of Rashford, and United's lack of experience was clear. The England international was introduced from the bench with Andreas Pereira by Solskjaer after 63 minutes, as Greenwood and James made way.
Rashford's arrival helped United move up a gear, and a quick breakaway allowed Mata to open the scoring. Martial fed the Spain international, who dinked the ball over Ruddy with a superb finish.
A heavy challenge saw Rashford's 17-minute cameo come to a premature end with 11 minutes remaining. The forward worryingly hobbled off straight down the tunnel.
Wolves responded to Rashford's exit and chased hard for an equaliser. United appeared nervous in midfield and defence as they tried to manage the closing stages.
However, the visitors failed to create at the death, and it's United who progress to play in the fourth round.