Arsenal failed to build on their early advantage, but the point is enough to take the Gunners above Chelsea into fourth position in the Premier League table.
Ryan was seriously called into action in the fifth minute, though, as he had to claw an inventive Aubameyang effort over from just under his own crossbar.
Arsenal did take the lead in the seventh minute, however, when Aubameyang picked out the top corner with a wonderful effort following good work from Lacazette.
Pascal Gross was next to try his luck for the home side in the 22nd minute, but the midfielder's effort was deflected wide of the post.
Glenn Murray did have the ball in the back of the Arsenal net shortly after, but the goal was disallowed for a foul on Leno.
Murray then tested the Arsenal goalkeeper with a low strike 15 minutes before the end of the first period as the home side continued to improve.
Aubameyang should have doubled Arsenal's lead in the 31st minute when he broke into the Brighton box, but the in-form forward saw his low effort brilliantly saved by Ryan.
Instead, it was Brighton that scored the game's second goal in the 35th minute. The equaliser came from an Arsenal corner, with the Seagulls launching a quick counter-attack which ended with Locadia poking the ball past the onrushing Leno before sliding it into the back of an empty net.
Propper then headed just wide of the Arsenal post in the 53rd minute as the away side threatened again.
The same player had another opportunity just before the hour when the ball broke kindly for the midfielder inside the Arsenal box, but he struck wide of the post.
That said, March had a wonderful chance to Brighton ahead in the 77th minute when he broke onto a long straight ball, but the attacker lifted his effort over the crossbar.
Locadia then struck just wide of the Arsenal post following smart work in a wide position.
Both teams had half-chances in the final minutes, but neither could score a second as the points were shared at the Amex Stadium on Boxing Day.
BRIGHTON (4-3-3): Ryan; Montoya, Duffy, Balogun, Bernardo; Stephens, Propper, Gross; March, Murray (Andone 79'), Locadia (Knockaert 86')
ARSENAL (4-3-3): Leno; Lichtsteiner, Sokratis, Koscielny (Maitland-Niles 70'), Kolasinac; Xhaka, Guendouzi, Torreira; Ozil (Iwobi 46'), Lacazette (Ramsey 62'), Aubameyang
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