Romelu Lukaku scored early in both halves at the John Smith's Stadium to secure the visitor's safe passage into the quarter-finals of the competition.
But the night once again saw technological teething problems on the controversial VAR system which is being specifically tested in this season's tournament.
Juan Mata wheeled off in celebration after netting from an acute angle just before half-time, only for referee Kevin Friend to eventually chalk the goal off as the narrowest of offsides was revisited.
United boss Mourinho was happy to see technology utilised and pointed to its relative success in Portugal, but felt it was wrong for video review assistant Neil Swarbrick to get involved given it was not a clear and obvious error.
"It is amazing when the truth comes," the United manager said.
"But also (there are) problems and one of the problems is exactly what happened today, which goes a little bit against the words of the protocol.
"The words of the protocol are something like a clear and obvious situation and for sure it was not a clear and obvious situation.
"But we know that it is experimental, and we have to accept. But I am speaking after a 2-0 victory."
Mourinho was then asked whether his view would be different had it cost United.
"I am close to winning the best-behaved manager in the Premier League, so nothing that can make me lose that trophy this season," he said, smiling once more.
"The feeling is a bit strange and I was sharing it with David (Wagner, the Huddersfield boss) on the touchline and (fourth official) Anthony Taylor, which is the moment of the doubt.
"We score a goal, I cannot be happy.
"David concedes a goal but he doesn't know if he goes to half-time with 2-0 or 1-0, and it takes time and then the decision comes.
"So, we have to adapt to that waiting time and cope with the frustration when that decision goes against you.
"At half-time I had to say 20 times to my analysts and assistants to shut up because I don't want the players to know what happened.
"I want the players to feel, 'Yes, it is offside and to move on', without being sad or unhappy.
"'Shut up, we have to move on', but one day it will go in our favour."
United's reward for a clinical performance in West Yorkshire is a home-tie against Brighton & Hove Albion in the last eight of the competition.
"I am happy that we are in the quarter-final," Mourinho added.
"I am happy we won a match, especially after the defeat (at Newcastle last weekend).
"It was the kind of game where a draw would not be a drama to take the match to Old Trafford, but the game was difficult.
"I think they had a different approach to what they had in the Premier League."