The Potters showed a lot of promise in the opening 15 minutes but they were unable to test David Ospina, who was deputising for the injured Petr Cech.
Arsenal, who have a Europa League quarter-final clash against CSKA Moscow on Thursday, eventually got going after 20 minutes and went close to scoring through Aaron Ramsey's lob.
Chances were few and far between in the first half, with tempo dying down in the final 10 minutes of the half.
Danny Welbeck had a quiet first half but he went close to scoring in the 55th minute with a volley that sailed just wide of the mark.
Both sides opted for a change on 60 minutes as Saido Berahino replaced Diouf for Stoke while Welbeck made way for Alexandre Lacazette.
Arsenal were awarded a penalty in the 74th minute when Ozil was brought down by Bruno Martins Indi. Pierre-Emerick Aubameyang scored the spot-kick to put the hosts ahead.
Aubameyang completed his brace in the 86th minute with a cool finish into the bottom corner to end any hopes of Stoke getting back into the game.
With one minute remaining, Arsenal were awarded a second penalty as Lacazette was fouled in the box. The striker took the spot-kick himself and sent Butland the wrong way.