Ozil was included from the start after shaking off a knee problem and was twice on target against the Canaries.
Olivier Giroud set up Jack Wilshere to open the scoring on 18 minutes with a classy flick in the penalty area following an outstanding team move.
After the break, a quick Arsenal counter-attack resulted in Giroud crossing perfectly from the right for Ozil to head home past John Ruddy and open his Premier League account.
Despite losing heavily and remaining in the relegation zone, the Canaries can take heart from aspects of their performance. They tested Wojciech Szczesny on several occasions and Jonny Howson reduced the deficit on 70 minutes with a low finish.
However, substitute Aaron Ramsey made sure of victory in the 83rd minute as he tricked Howson and Sebastien Bassong in the area before netting confidently.
And Ozil put further gloss on the scoreline when he tapped home from close range as Ramsey nudged the ball into the six-yard area following a deep Tomas Rosicky left-wing cross.
The result took the Gunners back to the top of the table after being briefly overtaken by Liverpool; they lead the Reds and Chelsea by two points after eight games.
Arsene Wenger made two changes, bringing the fit-again Santi Cazorla and Bacary Sagna back into the starting line-up. Ramsey and Carl Jenkinson dropped to the bench. Gary Hooper was included in place of the absent Ricky van Wolfswinkel (toe) for the visitors.
The hosts made a bright start and, after 18 minutes, Wilshere - who broke a league goal drought of almost three years when he equalised at West Brom last time out - put them ahead with his first Premier League goal at the Emirates.
An excellent team move instigated by Cazorla was at risk of breaking down when it reached the area due to the quick interchange of passes, but Giroud received the ball again between four Norwich defenders and produced a skilful flick through for Wilshere, who was just onside. The England midfielder coolly slotted home.
Ruddy denied Giroud with a brilliant reflex save at his near post following a run and cross down the left by Kieran Gibbs.
Mathieu Flamini had to go off through injury after 37 minutes following a collision with Alex Tettey, allowing Ramsey to be introduced. And as the half ended, Norwich came close to levelling as Szczesny was called upon to deny Leroy Fer who had struck a low effort from 20 yards.
The Canaries maintained pressure after the restart but were undone by a swift, slick Arsenal response just before the hour mark.
Moving quickly downfield, Ozil motored down the left channel off the ball and arrived with perfect timing to meet Giroud's lofted delivery with a well-executed header.
Cazorla made way for Rosicky, and Norwich got some reward for a committed display when Howson charged onto the end of a Per Mertesacker half-clearance and blasted the ball into the bottom corner past Szczesny.
Chris Hughton threw on Nathan Redmond and Wes Hoolahan in the hope of forcing an equaliser, while Nicklas Bendtner was brought on by Wenger.
However, Ramsey showed outstanding balance and close control to evade two attempted Norwich tackles in the area and extend Arsenal's lead, while Ozil wrapped up the win after the Welshman teed him up in the 88th minute.
The result will give Arsenal even more confidence ahead of Tuesday's Champions League visit of Borussia Dortmund, while Norwich move on to a home game against Cardiff next Saturday, when the Gunners travel to another of the promoted clubs, Crystal Palace.