Two first-half goals in the space of 90 seconds were enough for the Gunners as Mesut Ozil fired in from close range before Alex Oxlade-Chamberlain netted a fine effort.
The result keeps the Cherries five points above the Barclays Premier League drop-zone.
In a scrappy opening 20 minutes, Mathieu Flamini was booked for a lunge on Dan Gosling with two feet, with Kevin Friend producing a yellow card for the French midfielder despite vehement Cherries protests.
The game began to open up and the Gunners took full advantage to take the lead after 22 minutes.
Aaron Ramsey put in a cross from deep towards the far post and Olivier Giroud rose highest to cushion a header down for Ozil, and the German smashed a half volley in from eight-yards.
Just 90 seconds later, the Gunners had their second and Ramsey was instrumental again.
He played a ball through for Oxlade-Chamberlain and from a tight angle on the right side of the box, the winger drilled a low effort off the far left post and in - his first ever Barclays Premier League away goal for Arsenal.
The second goal seemed to spark Eddie Howe's men into life and Harry Arter went close shortly afterwards, he blasted a swerving effort from 20-yards which Petr Cech had to be at his best to tip over.
Just before the break, Cech was called upon again as a long ball found Adam Smith who ghosted past Nacho Monreal in the box and the visitors' stopper came out quickly the thwart Smith at his feet when one-on-one.
The Cherries came out on the front foot after the break and were on top, with Smith again going close just five minutes after the interval when he blasted a volley into the side netting in the box.
Artur Boruc then made a great save against the run of play to deny Alexis Sanchez's low strike from in the area, but the home side continued to press and probe the Arsenal backline.
Matt Ritchie and Dan Gosling were then replaced by Junior Stanislas and Joshua King as Eddie Howe looked to give the Cherries more impetus going forward.
But the Gunners were able to weather a brief Cherries storm and almost added a late third as Ramsey went through one-on-one but instead of passing to Giroud, he went for goal in the area but Boruc produced another excellent save to deny him.
Cech did have two last gasp saves to make, as he first denied Steve Cook's low effort from eight-yards before getting up to make another tremendous stop to get down low to deny Arter's vicious strike.
That was the last of the action though as the Cherries were beaten by the Barclays Premier League title hopefuls for the second time this campaign.
AFC Bournemouth: Boruc, Smith, Francis, Cook, Daniels, Surman, Gosling (King 65), Arter, Ritchie (Stanislas 65), Pugh, Afobe (Grabban 76).
Unused subs: Federici, Distin, O'Kane, Iturbe.
Arsenal: Cech, Bellerin, Gabriel, Koscielny, Monreal, Oxlade-Chamberlain (Coquelin 67), Ramsey, Flamini, Sanchez (Gibbs 81), Ozil, Giroud (Walcott 90).
Unused subs: Ospina, Mertesacker, Campbell, Iwobi.
Goals: Ozil (22), Oxlade-Chamberlain (24)
Attendance: 11,357 (1931 away fans)
Referee: Kevin Friend
Yellow card: Flamini