Gabriel Jesus' goal for Arsenal was his third in the Premier League this season
The Gunners, now a point clear at the summit, claimed their success thanks to Gabriel Jesus' header and Kai Havertz's late strike.
Mikel Arteta's side dominated the first half against Brighton without reward, but the tension eased when Jesus turned in after a corner.
The hosts were almost punished when Pascal Gross poked a golden chance wide before Havertz sealed victory to spark joyous celebrations among a relieved home contingent.
The Seagulls, who were looking for a third successive league victory at Emirates Stadium, managed just one shot on target and drop to ninth place.
After a conservative first-half performance, Brighton offered more attacking threat in the final half an hour but anything other than a home win would have been scarcely deserved.
Arsenal had 26 shots in the match and almost added a third in injury-time when substitute Emile Smith Rowe's low drive was saved.
"It was an incredible performance. It was a joy to watch from the start to the end," said Arsenal boss Arteta, whose jubilant celebrations indicated the magnitude of the win.
"The way we played against a phenomenal Brighton side was incredible. We were patient, kept going, kept believing and got what we deserved."
With Liverpool frustrated at home by United, Arsenal now have 39 points from 17 games and next head to Anfield on Saturday for a crucial meeting with Jurgen Klopp's side.
-- Courtesy of BBC Sport