The game began slowly in front of a record crowd at Brighton's new stadium before Olivier Giroud broke the deadlock on the quarter-hour mark with a curling shot.
Theo Walcott won the game in the 85th minute after coming off the bench.
Below, Sports Mole picks the bones out of a classic cup tie on the south coast.
On target 3
On target 6
Was the result fair?
Brighton will feel really aggrieved to have not earned a replay in North London after pushing their Premier League opponents all the way. Arsenal's quality up front was just about enough to book their place in the fifth round.
After a meaningless opening quarter of an hour, Brighton were hit on the counter by Giroud's sublime finish. The move came after Brighton had missed a golden chance down one end, with Wojciech Szczesny beating away Barnes's close-range volley. Gus Poyet's side responded well to going behind and equalised soon after when Leonardo Ulloa headed home on his debut. They again pegged Arsenal back after Giroud's second but were denied a replay because of a cruel deflection on Walcott's volley.
Good in parts. The performance of Olivier Giroud will please Wenger most but both goals conceded by the Gunners were down to slack marking. In the end though their quality paid off and they looked most likely to win it in the dying stages.
Sports Mole's man of the match
Olivier Giroud: No contest for this award today. The muscly Frenchman took both his goals quite brilliantly and he's now got 13 goals for the Gunners.
Has to be the performance of Andre Santos. The left-back has flopped completely since coming to England and Brighton took advantage of his lax defending, especially in the first half.
Michael Oliver did everything right today. The young referee was presented with no contentious decisions and dished out bookings where appropriate.
Brighton: The Seagulls return to Championship action at Sheffield Wednesday in a week's time.
Arsenal: Arsenal, meanwhile, welcome Liverpool to the Emirates for a Premier League game on Wednesday.