A 27th minute header from Gabriel gave the hosts the lead, before Mesut Ozil completed the scoring following a smart one-two in the second half.
Joshua King returned to the starting line-up for the trip to the Emirates, and the Norway international almost made an immediate impact.
Harry Arter twice slipped King through in the opening ten minutes, but Petr Cech was equal to the first curled effort before the former Manchester United man then ran the second into touch.
Marc Pugh was looking to make an early impression, too. The winger was wreaking havoc down the left flank and his half-cleared cross almost provided the opener, but Arter’s volley was well blocked.
The home side were next to try their luck, and a long ball over the top fell fortuitously to Theo Walcott inside the Cherries penalty area. But the Arsenal man mistimed his strike and Artur Boruc was on hand to reclaim the ball.
It was, however, the hosts that eventually broke the deadlock on the 27th minute. A whipped Ozil corner found the head of Gabriel who nodded past Boruc to give Arsenal the lead.
Moments later the home side came within inches of doubling their tally from the same position. Ozil’s cross was this time headed onto the post by Gabriel before Per Mertesacker put the rebound wide from close range.
Boruc then saved well just before the break, as Walcott looked to dink the ball over the Polish stopper to double his side’s advantage.
Eddie Howe made one change at the break, introducing the in-form Junior Stanislas onto the left flank in a straight swap for Pugh.
It was the Cherries that had the first major chance of the second half. King slipped the offside trap and bore down on the Arsenal goal, but the striker was pushed wide and had his eventual cross blocked.
With half an hour remaining Howe made his second substitution of the evening, replacing Gosling with Eunan O’Kane.
Arsenal then doubled their lead shortly after as Ozil played a quick one-two with Olivier Giroud on the edge of the penalty area, before the German then tucked the ball into Boruc’s far corner.
Despite the 2-0 score line, AFC Bournemouth continued to cause Arsenal problems and a passing move from the back resulted in Matt Ritchie trying his luck from the edge of the penalty area. But Cech safely clutched the Scotland international’s rasping drive.
With ten minutes remaining Howe made his final change of the evening, replacing King with summer signing Glenn Murray.
The Cherries fought right up until the final whistle with late attempts from both Stanislas and Murray, but neither could beat Cech and find the back of the Arsenal net.
Arsenal: Cech, Bellerin, Gabriel, Mertesacker, Gibbs (Monreal, 81), Chambers, Ramsey, Oxlade-Chamberlain (Iwobi, 90), Ozil, Walcott, Giroud (Campbell, 80).
Subs not used: Ospina, Debuchy, Koscielny, Reine-Adelaide.
Booked: Ramsey, Chambers.
Goals: Gabriel, 27, Ozil, 63.
Cherries: Boruc, Daniels, Cook, Francis, Smith, Gosling (O'Kane, 62), Arter, Surman, Ritchie, Pugh (Stanislas, 45), King (Murray, 80).
Subs not used: Federici, Distin, Tomlin, Rantie.