Moments after almost slicing the ball into his own net, Lewis Dunk scored his first PL goal with a close-range finish on seven minutes.
Anthony Knockaert had an effort well saved by Petr Cech before Glenn Murray headed in Brighton's second from a superb Pascal Gross cross.
Pierre-Emerick Aubameyang pulled a goal back towards the end of the first half, with a deft flick from Granit Xhaka's pass, before Arsenal defender Laurent Koscielny hit the post.
Mesut Ozil and Aubameyang went close for Arsenal while Murray had a goal ruled out for offside, but Brighton held on to move seven points clear of the relegation zone. Arsenal remain sixth.
Brighton: 10 Mar v Everton (A)
Arsenal: 11 Mar v Watford (H)