Nacho Monreal and Alex Iwobi netted either side of half time as the Gunners registered their fourth league clean sheet in a row and equalled a club record of nine successive wins at home dating back to March.
Brighton gave a good account of themselves in North London, but they remain without an away win in the top flight since gaining promotion in May.
Within 70 seconds of kickoff, Arsenal were almost in front through Alexandre Lacazette, who cannoned the ball off the post from 25 yards out.
Alexis Sanchez also tried his luck from distance, although Mathew Ryan was equal to his 30-yard strike, before Iwobi was guilty of placing the ball too close to the Australian goalkeeper from inside the penalty area.
However, the Gunners did take the lead in the 16th minute when Monreal found the net from around 12 yards out after Brighton had earlier cleared off the line during a scramble inside the box.
Despite the scoreline, Brighton were not being outclassed, with Isaiah Brown having earlier glanced a header wide of the post, but the Seagulls soon came even closer to getting on the scoresheet.
However, as the first half approached a conclusion, Arsenal began to assert themselves on the game, but squandered two clear-cut openings to double their advantage.
After a slip in the Brighton defence, Lacazette powered forward but after deciding against passing to a teammate, the Frenchman curled the ball too close to Ryan.
Shortly before the break, Arsenal put together a rapid counter-attack which saw Hector Bellerin and Granit Xhaka combine to set up Aaron Ramsey, but the Welshman was somehow denied by Ryan from just eight yards out.
After the restart, Arsenal took their time before getting back into their stride, but a second goal was inevitable and it soon arrived from Iwobi.
Sanchez was the creator, with his back-heel going straight into the path of Iwobi, who bided his time before finding the top corner from the right-hand side of the box.
Arsenal wasted no time in trying to net a third, with Sanchez the next player to be denied by a sensational stop from Ryan, who got down to his left to tip the Chilean's low shot behind for a corner.
From the resulting set piece, Kolasinac rose above the Brighton defence and beat Ryan with his header, but his effort was cleared off the line by a defender.
A wave of substitutions disrupted the rhythm which Arsenal had built and as the game entered the closing stages, Brighton found a second wind to put their hosts under sustained pressure.
Centre-back Mustafi then glanced the ball marginally wide of his own goal as Brighton began to lay siege to Cech's goal in search of a goal which would only increase the nervousness in the Arsenal defensive third.
In added-on time, Davy Propper curled agonisingly wide of the far post from 20 yards and that proved to be Brighton's last chance as Arsenal held on for yet another shutout in domestic action.