Antonio Valencia and Lingard pounced on defensive mistakes by Laurent Koscielny and Shkodran Mustafi as Jose Mourinho's visitors took a 2-0 lead inside 11 minutes at the Emirates Stadium.
Arsenal refused to buckle. Alexandre Lacazette netted when United's defence stood still, but De Gea made a string of fine saves to deny the Gunners.
Lingard struck on the counter-attack following fine work from Pogba, who was then shown a straight red card for a studs-up challenge on Hector Bellerin.
The midfielder's offence carries an automatic three-match ban and rules Pogba out of next weekend's Manchester derby.
The win saw second-placed United move five points behind rivals City, the Premier League leaders playing West Ham on Sunday, while Arsenal are seven points further adrift in fifth.