Goals from Lewis Dunk and Glenn Murray gave Brighton a two-goal advantage and although Pierre-Emerick Aubameyang halved the deficit before half time, Arsenal were unable to force an equaliser after the break.
The Gunners - who have now suffered four successive defeats in all competitions - remain eight points adrift of fifth-placed Chelsea, but Brighton have moved into the top half of the standings with their third top-flight win in a row.
Arsenal began brightly on the South Coast with Mesut Ozil curling marginally over the crossbar from 20 yards out, but Brighton soon began to control proceedings in front of their own supporters.
Brighton's opener came from a set piece, with an inswinging corner from the left finding Shane Duffy - who beat Petr Cech in the air - and the ball was knocked down for Dunk to fire a shot into the roof of the net on the half-volley.
Moments later, Pascal Gross forced Cech into a sharp save to his right, before Dunk wasted an opportunity to net his second in the first quarter of the match as the centre-back sent a free header over the crossbar.
Anthony Knockaert also squandered a chance after being picked out by Gross's backheel, only for the winger to be denied from 12 yards by a block from Cech, but the veteran goalkeeper was at fault for Brighton's second goal.
After Laurent Koscielny had lost possession, Gross crossed for Murray who was waiting at the back post and the forward found the net from 10 yards after his header went under the body of the Czech stopper.
Brighton remained in the ascendancy but with two minutes remaining until half time, Arsenal found a way back into the game when Granit Xhaka's pass inside the penalty area picked out Aubameyang, who flicked the ball past Mathew Ryan from seven yards.
Just before the break, the Gunners almost found an undeserved equaliser when Koscielny's header was deflected onto the post, but Brighton were able to scramble the ball away before reaching the interval with their advantage still intact.
After the restart, Arsenal were in the ascendancy and Henrikh Mkhitaryan soon got a shot in on goal, with Ryan comfortably dealing with his shot from 20 yards.
Brighton were still offering a threat in the final third but when the home side did get the ball into advanced positions, they were not showing the same level of composure as in the first half.
Just before the hour mark, Ozil played a one-two with Aubameyang before seeing his 25-yard strike tipped behind for a corner by Ryan, who was becoming increasingly busy between the sticks.
However, the next chance fell to Brighton's Jose Izquierdo, who was allowed to drift inside onto his right foot before sending a low shot marginally wide of Cech's right post.
As the match entered the final quarter, Wenger introduced Danny Welbeck but while Arsenal started to throw more men forward, Brighton were sitting deeper and showing signs that they were prepared to play on the break for the remainder of the match.
The Gunners dominated possession during the closing stages of the game but their one clear opening fell to Aubameyang, who could only send a half-volley straight at Ryan from 12 yards out.
Brighton had their own openings during seven minutes of added-on time, but Chris Hughton watched on as his side successfully held on to their slender lead to register their first points over a top-six side this season.
BRIGHTON & HOVE ALBION (4-5-1): Ryan; Scheletto (Bruno 69'), Duffy, Dunk, Bong; Stephens, Propper, Knockaert (March 77'), Izquierdo, Gross (Kayal 87'); Murray
ARSENAL (4-2-3-1): Cech; Chambers (Bellerin 83'), Koscielny, Mustafi, Kolasinac; Wilshere, Xhaka, Iwobi (Welbeck 74'), Mkhitaryan (Nketiah 83'), Ozil; Aubameyang