Ozil struck with a tremendous effort in the 24th minute, and it proved enough for Arsenal to end their three-game winless streak and register a 10th successive victory over Newcastle in the process.
The Magpies had their moments at the Emirates Stadium but, despite a strong second-half showing, they have now failed to win any of their last nine matches in the top flight.
The first shot of the match came from the recalled Alexandre Lacazette, who drilled an effort straight at Rob Elliot from distance, but Newcastle also had some joy during the early stages as Jamaal Lascelles saw a header diverted wide by Petr Cech.
However, Arsenal soon began to control the game in North London and, just after the quarter-hour mark, Alexis Sanchez sent a shot over the crossbar after being teed up by Ozil.
An opening goal had been coming and it arrived from Ozil, who capitalised on a half-clearance from Newcastle to send a volley into the top corner of the net from the edge of the area.
Hector Bellerin was providing a constant threat down the right flank but it was Ozil who should have doubled his team's lead just before the break, with Newcastle being grateful for Elliot making a stunning save to keep out the midfielder's effort from close range.
A similar pattern emerged after the restart but, aside from Christian Atsu almost putting through his own net, Arsenal were struggling to force Elliot into more work between the sticks.
However, Arsenal should have wrapped up the points soon afterwards when the ball fell to Alex Iwobi from no more than eight yards out, but Elliot made yet another superb save to keep his side in the match.
Bellerin had a chance to wrap up the points for Arsenal in the closing stages but despite Elliot adding to his highlight reel of saves, Arsenal were able to get over the line to record a much-needed three points.
ARSENAL (4-3-3): Cech; Bellerin, Koscielny, Monreal, Maitland-Niles; Xhaka, Wilshere, Iwobi (Welbeck 72'); Ozil, Sanchez (Coquelin 89'), Lacazette (Giroud 73')
NEWCASTLE UNITED (4-5-1): Elliot; Yedlin, Lejeune, Lascelles, Manquillo; Hayden, Merino (Diame 82'); Atsu (Ritchie 54'), Perez, Murphy (Gayle 67'); Joselu