Ozil, signed from Real Madrid on deadline day, had a hand in all three of Arsenal's goals as Arsene Wenger's men finally ended the resistance of Mark Hughes' well-organised team.
Aaron Ramsey smashed in his seventh goal of an impressive season so far after Ozil's fifth-minute free-kick was parried by Asmir Begovic, but Geoff Cameron swept in a deserved equaliser for the visitors.
Ozil's class, though, proved the difference again as his corner was flicked home by compatriot Per Mertesacker to give Arsenal the lead again before half-time and another pin-point deadball delivery made sure of all three points when Bacary Sagna looped in a header on 72 minutes.
Arsenal have responded with seven straight wins since they were beaten 3-1 at home by Aston Villa on the opening day of the season - and if Ozil continues his impressive displays, Wenger's squad could just be set for a long-overdue sustained assault on the Premier League title.
Wenger was forced into a late change to his starting XI when England forward Theo Walcott suffered a minor abdominal strain during the warm-up and was replaced by youngster Serge Gnabry.
Denmark striker Nicklas Bendtner was on the bench for the first time this season, having seen a move away during the transfer window blocked.
It did not take long for Ozil to have an impact, as his 20-yard curling free-kick was parried out by Begovic - and Ramsey was first to the loose ball to slot home.
Ramsey wheeled away in delight, his finger to his lips in front of the Stoke fans, some of whom are always quick to remind the Wales midfielder of a double-leg break suffered in a tackle from Potters defender Ryan Shawcross at the Britannia Stadium in February 2010.
For his part Shawcross continued to be roundly booed by sections of the home crowd every time he was on the ball.
Source: PA
Source: PA