A 1-1 draw at home to Liverpool, after Philippe Coutinho's goal just before half-time cancelled out Anthony Martial's penalty, put the visitors through 3-1 on aggregate.
United created a number of chances in the first half in particular but lacked a cutting edge with goalkeeper David de Gea once again their best player.
It increases the pressure on Van Gaal, whose side are four points behind fourth-placed Manchester City with a trip to the Etihad Stadium next up, but he believes a derby is the one thing which can spark life into their bid for Champions League qualification.
"I am not angry, I am not frustrated
I was very proud of my players," said Van Gaal despite the pressure increasing on him.
"They gave everything and I was very pleased the fans recognised that
They applauded after the match despite us being out - that was remarkable I think.
"We have created a lot of chances; we scored a penalty, [chances for] Lingard, Mata, Fellaini, Martial, and then you concede a goal in the last seconds.
"That is what we did not do in Liverpool, they scored an away goal and that makes a difference how they feel in the dressing room at half-time.
"In the second half we tried to raise ourselves out of disappointment but you know, the fans know, everyone knows we have to score three goals and that was more difficult.
"There is always pressure
Expectations are very high so pressure for these players is always very high
We can cope with that but we were not lucky today.
"I hope City is the catalyst
We have to beat City and we have a chance to still qualify (for the Champions League)."
Liverpool boss Jurgen Klopp hailed a "genius moment" from Coutinho to finally beat de Gea, who prevented the result being much worse for United over the two legs.
"They got a penalty - it was a penalty - and Phil had a genius moment," said the German.
"I love it two minutes before half-time because he twice reacted quicker than his opponent.
"It was the most unexpected that he could do in a situation like this."
Klopp had wanted United before he was drawn against them but when asked whether he would now like to face former club Borussia Dortmund, the favourites, in the quarter-finals his view was different.
"Why should I want the strongest team in the next round - I'm not so silly?" he added.
"I saw only a bit of the other results but I am pretty sure there are no easy games so we have to be prepared."
Source : PA
Source: PA