England winger Walcott set the tone for a dominant first-half display from the holders with the opener in just the second minute, while Ozil - making his first start since October - added the second before half-time.
Having offered precious little in terms of an attacking threat prior to the break, Brighton reduced the arrears early in a much-improved second-half showing when Chris O'Grady capitalised on some poor Arsenal defending to score.
Hopes of a stirring comeback on a weekend of upsets appeared to be dashed when Tomas Rosicky volleyed in Olivier Giroud's chipped pass from the edge of the box.
There was still time for Sam Baldock to set up a nervy finish with a measured 75th-minute finish, but Arsene Wenger's men held firm.
