The Red Devils secured their passage to the last eight with a narrow 2-1 aggregate win over FC Rostov on Thursday.
United had been held in Russia but booked their place in Friday's draw in Nyon courtesy of Juan Mata's second-half strike, with Mourinho's men claiming a 1-0 victory at Old Trafford on the night.
And they will now face Anderlecht, who defeated APOEL 2-0 on aggregate, in the quarters, with the first leg set to take place in Belgium on April 13 although official dates are subject to confirmation.
The Premier League side have happy memories when it comes to facing their opponents having recorded their biggest ever European win over Anderlecht, a 10 win back in 1956.
Star striker Zlatan Ibrahimovic will also relish coming up against the side who finished third in the Jupiler Pro League last season having scored four against them for PSG in 2013.
The matches will only to add Mourinho's headaches with regards to fixture congestion, however, with United now scheduled to play eight matches in April.
The first leg is set to take place four days after a trip to Sunderland in the Premier League, with the second leg coming after the meeting with Chelsea, which is scheduled for April 16.
Mourinho complained about fixture congestion in the aftermath of the win over Rostov, insisting that United's Premier League rivals "don't know what fatigue is" when asked about Manchester City's Champions League exit.
He told BT Sport: "I am very disappointed [at City going out]. I am not making fun of it. It's bad for us for many reasons.
"It's bad for us as an English club and it's bad for us as all of the five teams in front of us in the Premier League don't have international football.
"They will be playing one match a week, like some of them have been doing since the beginning of the season.
"They don't know what fatigue is, they don't know what accumulation is, they don't know what it is like to go home and not sleep because of fatigue.
"We will fight to our limits - that's what we have to do."