Ozil was making his first appearance at north London club Arsenal's home ground following his £42.4 million ($66 million, 50 million euros) move from Real Madrid.
He made an immediate impression by setting up first-half goals for Aaron Ramsey and Per Mertesacker before delivering the free-kick that led to Bacary Sagna's match-clinching third goal.
Stoke cancelled out Ramsey's sixth-minute opener with an effort from Geoff Cameron but were undone on three occasions from set-pieces.
Victory moved Arsenal two points clear at the top after a fourth straight league win.
Ozil -- who had marked his first Gunners start with an assist after just 11 minutes at Sunderland last weekend -- quickly made himself at home, although Ramsey's goal owed more to a mistake by Stoke keeper Asmir Begovic than the German's free-kick.
The playmaker drilled in a low shot after Jack Wilshere's forward run had been halted by Charlie Adam and Begovic parried the effort tamely into the path of Ramsey who scored from close range.
The Wales international has now scored seven goals in seven Arsenal appearances and, while Ozil may have grabbed most of the recent attention, Ramsey's contribution has been equally eye-catching.
It was the ideal start for Arsenal after French manager Wenger had been forced to make a last minute change to his line-up when Theo Walcott suffered a recurrence of an abdominal problem in the warm-up.
German teenager Serge Gnabry stepped up from the bench and while the youngster's impact was limited, Arsenal initially coped well with the loss of Walcott.
Stoke though, overcame the early setback and began to impose themselves on the game.
They offered signs of their growing threat when Sagna was within inches of conceding a penalty when he tripped Adam in the 24th minute.
And two minutes later, the visitors were level after a well-worked move capped off by Cameron's volleyed finish.
Steven Nzonzi clipped a ball towards Marko Arnautovic who got the wrong side of Mertesacker before volleying against the post and Cameron converted the rebound from the edge of the area.
Arsenal's promising start had quickly faded and Ozil's influence had waned.
But Wenger's side soon rallied, re-establishing their lead ten minutes later after exposing Stoke's unexpected vulnerability in the air.
Mertesacker was unlucky to see his near post header from Ozil's corner blocked by an alert clearance by Stoke left back Erik Pieters in the 32nd minute.
But the home side repeated the same routine four minutes later with more success when Mertesacker's header looped inside the far post.
Stoke's ability at set-pieces has been one of the club's strengths in recent years but on this occasion they were overwhelmed by Arsenal -- and worse was to come in the second half.
Buoyed by the second goal, the Gunners began playing with confidence and freedom, summed up by the slick interchange of passes that almost brought a third goal for Olivier Giroud.
Stoke had looked compact during the opening half hour but holes began to appear in the visitors' defences and these were quickly exploited by Ozil and Wilshere, who are already showing signs of a intuitive understanding.
A rapid break at the start of the second half ended with Giroud forcing a fine save from Begovic and suggested the second period would follow the same pattern as the first.
Stoke's Marc Wilson tested Arsenal keeper Wojciech Szczesny but the visitors offered little going forward before they were undone by yet another free-kick in the 73rd minute as Sagna killed off the game.
Ozil was again the provider, this time clipping in a deep free-kick towards the far post where Sagna out-jumped Wilson and Jonathan Walters to send a header over Begovic and into the far corner.