Marouane Fellaini's early goal looked to have settled United's nerves and secured passage to the Stockholm showpiece, but Facundo Roncaglia's equaliser five minutes from time sparked a tense ending at Old Trafford.
The defender, along with Eric Bailly, was then dismissed a minute later as the two sets of players clashed in the middle of the pitch.
And John Guidetti failed to finish United off with an open goal at his mercy with the last kick of the game and Mourinho accepted that his side were made to suffer for not capitalising on their superiority in the first leg.
"We were the best team in the first leg but we never kill [the game off]," he told BT Sport. "We never score goals related to the chances we have. It was an open game at home, all the pressure on our side.
"They were completely free of responsibility and gave us a very hard match. We suffered until the end and it was open until the last second. But the boys gave everything they had. I'm really pleased for them.
"Now, after 14 matches, we are in the final. If we win the Europa League I am more than happy. It would be amazing."
Bailly's sending off will leave a sour taste, however, as he is now set to miss the final in Stockholm on May 24 after he raised his hand to Guidetti's face.
Mourinho thinks the 23-year-old might have been naive to get involved, particularly with Roncaglia around.
"I didn't see the red card incident but Eric Bailly was phenomenal," he said. "The game was emotional for everyone. Some kept control better than others.
"Facundo Roncaglia is not new in these situations, Bailly was probably naive. We now lose a very important player for the final, and we don't have many."
Mourinho also said the desire fuelled by his description of this as the most important game in the history of the club was pivotal to their progression to the final.
Afterwards Mourinho conceded the game was far from the biggest in United's history, but says the final in Stockholm with Ajax provides them with the chance to end an at times trying season in the ideal manner by sealing UEFA Champions League qualification.
Asked about the significance of reaching the showpiece, Mourinho said: "It means an opportunity to win a trophy and the opportunity to be back to the Champions League. It means the opportunity to end the season in a perfect way.
"I said that [it was their biggest game ever] but obviously Manchester United had other semi-finals, Champions League finals. Obviously this was not the most important match in the history of the club or myself. It was just a way to approach things.
"For Celta, for Vigo, it was. It was very important for us that we equal that desire.
"In the end, when the game was difficult, when they were better than us, when we were in trouble and not playing well, I think that mentality, desire, togetherness made the boys fight until the end."