Athletico Madrid managed to hang on to a 1-0 lead, which came in the 41st minute from a header by Brazilian Renan Lodi.
Coach Diego Simeone lauded Lodi for taking his team into the quarter-finals.
“Lodi's doing really well. He had to wait a long time [to get in the side], but football is marvellous because when you train hard and are ready to go, there's always a chance for you."
Manchester United manager Ralf Rangnick criticised referees for allowing the visitors to get away with time-wasting tactics in the second half, saying, "there was always someone lying on the floor".
He called it a "joke" that were four minutes of stoppage time.
With their elimination from the European competition, Manchester United now only have the English Premier League, where they lie fifth, in their search for silverware.