The Spanish defender drilled in the opener on 16 minutes after Hector Bellerin's blocked shot went in to his path.
Solly March hit a post with a ferocious free-kick as Brighton responded brightly before Aaron Ramsey went close.
Alex Iwobi doubled Arsenal's advantage after the break, when he found the top corner following a delightful backheel from Alexis Sanchez.
Arsenal are up to fifth following their fourth win out of four home matches this season. Brighton are 14th.