The visitors will leave north London perhaps feeling hard done by that they didn't take a point having made the Gunners sweat before Alexis Sanchez made sure of the result as the game headed into second half injury time.
It was the hosts that started the brighter and deservedly took the lead on 12 minutes after capitalising on a Steve Cook howler. After receiving the ball out from goalkeeper Adam Federici the Cherries defender tried to return the pass but woefully under-hit it, allowing Sanchez to pounce on the ball, rounding Federici and slotting home into an empty net.
Bournemouth rebounded and levelled the scores on 23 minutes from the penalty spot. Callum Wilson was lucky in most people's eyes to have been awarded the spot kick as he fell to ground rather easily against Nacho Monreal.
Referee Mike Jones pointed to the spot and Wilson sent Petr Cech the wrong way from twelve yards. The visitors should have taken the lead three minutes later but Adam Smith put his header over the bar while unmarked in the area.
Arsenal had their chances too and retook the lead on 53 minutes with Theo Walcott heading home at the back post from close range after Monreal had played across a fantastic ball to find his team mate.
Bournemouth didn't give up and had their moments to level at what was becoming a nervy Emirates Stadium, including on 79 minutes when Benik Afobe was denied by a brilliant close range save from Cech.
Arsenal eventually put the game to bed in stoppage time when a flowing move was finished off as Oliver Giroud squared the ball to Sanchez, who made no mistake with the finish.