After opening defeats to Manchester City and Chelsea, Arsenal has found its groove under Unai Emery against lesser opposition and survived a poor first half to see off winless Newcastle.
Xhaka opened the scoring in stunning fashion, beating Martin Dubravka from 25 yards, and he played a part in the second as Ozil effectively sealed the points.
Newcastle's heads dropped thereafter and, despite Ciaran Clark's stoppage-time header, the Magpies have just a solitary point from a daunting start to the season, while Arsenal's bid for a return to the top four is gathering steam.
Goalkeeper Petr Cech and the Arsenal backline looked shaky with the ball at their feet and continually invited pressure in the early stages, and he was required to beat away a Jacob Murphy diving header at the far post.
Lucas Torreira was brought on for Matteo Guendouzi for the start of the second half while Newcastle was forced into a change at the back with injured captain Jamaal Lascelles replaced by Clark.
Three minutes into the second half that reworked back four was breached as Federico Fernandez fouled Pierre-Emerick Aubameyang and Xhaka unleashed a wonderful curling free-kick that left Dubravka helpless.
Arsenal moved into complete command nine minutes later when Ozil's low finish beat the hand of Dubravka after Alexandre Lacazette's initial effort from Xhaka's left-wing pull back was blocked.
Aubameyang then fired narrowly wide at the far post but his wastefulness ultimately went unpunished as Arsenal saw the game out after Clark's header from Fernandez's cross.
Arsenal welcomes Ukrainian side Vorskla Poltava to Emirates Stadium in its opening UEFA Europa League clash before hosting Everton in the Premier League next weekend.
Newcastle, meanwhile visits a Crystal Palace side fresh from a win against Huddersfield Town, but still searching for a first home point this season.