De Gea denied Steven N’Zonzi and Luis Muriel from point blank ranges just before the end of the first half.
United, who are looking to reach the quarter-finals in their first appearance in the knockout stage of the Champions League since 2014, took a cautious approach despite boasting Alexis Sanchez, Romelu Lukaku and Juan Mata in their attack.
The Spanish side made it eight times on target compared to United’s one.
“The only moment where we felt relieved was in the last couple of moments of the first half when from a couple of our mistakes in possession they had a couple of good situations and De Gea made good saves,” United manager Jose Mourinho said.
He added that the result was neither good nor bad.
“Apart from that the game was even, but the statistics are what they are. I really feel the result reflected what the game was -- you ask me was it a good result, and I say it was not good and not bad.”
Donetsk bounce back against Roma
Shakhtar Donetsk recovered from a goal down to beat Roma 2-1 in the other Champions League tie on Wednesday.
Cengiz celebrated his Champions League debut for Roma with a 41st-minute goal for the visitors who are looking to reach the quarter finals for the first time in 10 years.
However, goals by Facundo Ferreyra and Fred in the second half helped Donetsk take lead ahead of the return tie at the Stadio Olimpico in three weeks' time.