Jesse Lingard scored twice after Antonio Valencia's early opener as second-placed United recorded back-to-back away wins in the Premier League for the first time this season.
However, the visitors needed De Gea to repel an Arsenal onslaught either side of half-time, with United's last line of defence making a number of eye-catching saves en route to a record-tying evening .
While Alexandre Lacazette did score early in the second half, Mourinho admitted his keeper's heroics had helped decide a "phenomenal game" between the long-time rivals at the Emirates Stadium.
"I told him after the match – what I saw today was the best goalkeeper in the world," the Portuguese told BT Sport.
"In big matches you need goalkeepers to be there for you, but all the effort from the players was magnificent.
"It was phenomenal game. I loved the way my team played and fought. But I have to also say that Arsenal played, in some periods, amazing, attacking football, creating difficulties for us.
"The plan was when they have to the ball we all have to defend, starting with the attacking players. When we have the ball, however, we had to counter-attack fast to score goals.
"Of course, they can say they had chances, but we fought and had a great goalkeeper, too."
8 – Man Utd have won more away games at Arsenal in the Premier League than any other side. Brag.— OptaJoe (@OptaJoe) December 2, 2017
The only disappointment for United was the second-half dismissal of Paul Pogba, with the Frenchman's red card for a stamp on Hector Bellerin ruling him out of next Sunday's crucial derby with Manchester City at Old Trafford .
Mourinho refused to be drawn on the incident involving Pogba.
"I don't know. I leave for you [the media] that one," he commented.
"I'll leave for you the Lukaku situation with Koscielny and I'll leave for you the Arsenal players on the grass but I'll probably [think] the same because the grass is absolutely beautiful.
"I think there is a desire to go for the grass!"