The draw leaves Unai Emery's men in fifth spot, three points behind north London rivals Tottenham, who have a far superior goal difference, and means they can no longer catch Chelsea, who have 71 points.
Leaders Liverpool and Manchester City have already sealed the top two spots ahead of the final round of Premier League fixtures next weekend.
Arsenal can also reach the Champions League by winning the Europa League and they are in a strong position to reach the final, 3-1 up after the home leg of their semi-final against Valencia.
Emery made five changes for their final Premier League home game, juggling the club's league ambitions with Thursday's second leg in Valencia, including recalls for Nacho Monreal, Lucas Torreira and Henrikh Mkhitaryan.
Arsenal bid to reverse slump as Brighton fight for survival
Arsenal were impressive in the early stages at the Emirates and Pierre-Emerick Aubameyang struck from the penalty spot in the ninth minute after Monreal was brought down in the area by Alireza Jahanbakhsh.
The visitors, whose Premier League survival was assured when Cardiff lost at home to Crystal Palace on Saturday, came back strongly, putting pressure on a nervy home side but Aubameyang went close shortly before half-time.
Brighton levelled from a disputed penalty in the 61st minute.
Solly March went down under minimal contact from Granit Xhaka and Glenn Murray found the net from the resulting spot-kick.
Aubameyang missed a golden chance to restore Arsenal's lead as the home side poured forward and Ryan blocked an effort from Alexandre Lacazette.
Emery then made a triple substitution, throwing on Matteo Guendouzi, Alex Iwobi and Sead Kolasinac for Mkhitaryan, Xhaka and Lichtsteiner but despite frantic late Arsenal pressure, they could not find a winner.
They have a goal difference of +20, compared to Tottenham's +28, meaning Spurs, who host Everton next week, are almost assured of finishing the season in fourth spot.
Emery cautious as Lacazette, Aubameyang edge Arsenal towards Europa League final
Arsenal travel to face Burnley in their final league game.
COMMENTS
Comments are moderated and generally will be posted if they are on-topic and not abusive.
For more information, please see our Comments FAQ