De Gea made 14 saves, equalling the record for the most stops in a Premier League match.
"I'm just disappointed and angry that with the quality of that performance we have nothing to show for it", he said, attributing his team's loss to a lack of efficiency in finishing and De Gea's "outstanding game".
Discussing the Pogba red card, he said: "I don't know".
Jose Mourinho revealed Paul Pogba was furious with Arsenal's Laurent Koscielny for his role in the red card that rules the Manchester United midfielder out of next weekend's summit meeting with Manchester City.
Danny Welbeck and Alexandre Lacazette, appealed in vain for penalties and, with his tongue firmly in his cheek, Mourinho said: "I know the pitch is unbelievable and maybe breeds a desire to go to the floor".More news: Nathan Redmond Refutes Report That Pep Guardiola Called Him a 'W--ker'
City are five points ahead of United, but can extend that lead to eight if they beat lowly West Ham on Sunday. In my case, it's different because I want to win.
"We are second and are chasing the top of the table".
Despite the strength of United's counter-attacking and defensive display, the brunt of the attention was fixated on Arsenal's failure to capitalise on their consistent domination and understandably, the fans collectively voiced their frustrations with their team's result, despite the strong performance they put in over the ninety minutes.
"I told him after the match, what I saw today was the best from a goalkeeper in the world".
The scoreline flattered United with Arsenal playing with verve and pummeling De Gea's goal.
In a post-match huddle which also included Jose Mourinho, Thierry Henry and William Gallas, he gestured that it may have been Bellerin's unusual attempt to tackle that was to blame.More news: Former Yemen rebel ally Saleh calls for talks with Saudi Arabia
"Last night I watched Napoli v Juventus on TV and we watched a top-class referee".
"Of course, they can say they had chances but we fought and we had a great goalkeeper too and I think we deserved it".
It was off a bad giveaway, and Arsenal had a second one moments later.
"The concentration at the back wasn't sharp enough at the start", Arsenal manager Arsene Wenger said.More news: Dana White Unsure if Conor McGregor Will Ever Fight Again