Unai Emery’s side had taken an early lead through Pierre-Emerick Aubameyang’s fine first half strike.
But Locadia levelled for Brighton before half-time and the Gunners couldn’t find a winner after the interval.
Emery started Aubameyang and Alexandre Lacazette up front and the pair were quickly into the groove.
Despite not often starting together as central strikers this term, Aubameyang and Lacazette have established a good understanding and both were denied well by goalkeeper Mat Ryan early on.
Even so, it took just seven minutes for the pair to put Arsenal ahead. Lacazette poked through for Aubameyang, who curled a fine effort high into the far corner for his 15th goal of the season.
Brighton, beaten in their previous three games, thought they had equalised, only to see Glenn Murray’s effort ruled out.
Murray bundled his way towards goal and forced the ball, seemingly, over the line.
But it was adjudged to have been kicked out of the hands of Arsenal keeper Bernd Leno, who still scrambled back to attempt to claw the ball off the line.
Throughout their impressive first season under Emery, Arsenal have remained troubled by the defensive deficiencies that hampered them in final years of the Arsene Wenger era.
The Swiss international attempted to head a long ball back to Leno, but succeeded only in finding Jurgen Locadia, who took the ball past Leno and tapped into an empty net.
Emery made a half-time change, bringing on Alex Iwobi for Mesut Ozil, increasing the feeling that the out-of-form German midfielder is on borrowed time at Arsenal, where he is the highest-paid player.
Brighton began the second period the stronger, twice coming close to taking the lead.
Solly March cut in from the right and his low effort was pushed around the post by Leno. Then March turned provider to tee up Locadia who glanced just wide.
Iwobi almost won it late on when he blocked March’s clearance and the ball trickled just wide, but Arsenal were force to settle for a point.