In a dramatic game in south Wales, Championship side Swansea were 2-0 up against Premier League leaders City shortly before the half hour through a Matt Grimes penalty and fine strike by Bersant Celina.
There was also debate over City's equaliser, which came from the penalty spot after Cameron Carter-Vickers was adjudged to have fouled Raheem Sterling. "When we analyse the game we deserved to win by far".
Manchester City's torrid form continues with the win, now sitting first in the Premier League table and having demolished Schalke 7-0 midweek to reach the Champions League quarterfinals.
Sergio Aguero was then in an offside position when he scored the victor in the 88th minute, but the Video Assistant Referee was not in use as the game was not being played at a Premier League ground.
"We are in the dark ourselves over the non-use of VAR here and that it would only be used in Premier League stadiums", the spokesman told Press Association Sport.
From then on in it was all Manchester City and an own goal off the back of Swansea goalkeeper Kristoffer Nordfeldt after substitute Sergio Aguero's 77th-minute penalty had come back off the post leveled the tie.
Raul Jimenez and Diogo Jota sent Wolves into the semi-finals of the FA Cup for the first time in 21 years with two goals in six second-half minutes, with United only having Marcus Rashford's late consolation to show for their efforts.
In the early game on Saturday Andre Gray came off the bench to fire Watford into last four with a late victor in their 2-1 victory over Crystal Palace.
They are top of the Premier League, in the Champions League quarter-finals and have already won the League Cup.