Ozil was the centre of attention at the Stadium of Light following his £42.5million switch to the Gunners from Real Madrid, and he was quickly into his stride to set up Olivier Giroud for an exquisite early goal after just 11 minutes.
Arsenal carved out numerous chances to extend their lead before the interval as Giroud, Ozil and Jack Wilshere terrorised the Sunderland defence, but Theo Walcott was denied by Keiren Westwood on three occasions and headed another glorious opportunity wide.
Arsene Wenger's men were made to pay for those missed openings just three minutes into the second half as Laurent Koscielny needlessly brought down Adam Johnson in the area and substitute Craig Gardner stepped up to equalise from the penalty spot.
Sunderland kept the pressure on but it was Arsenal who managed to restore their lead in the 67th minute as Ramsey struck a crisp volley past Westwood from Carl Jenkinson's right-wing cross.
Jozy Altidore had the ball in the net for Sunderland but his effort was controversially disallowed and, after Ramsey rounded off another flowing move on 78 minutes to seal the points for Arsenal, Black Cats boss Paolo Di Canio was banished from the touchline in the closing stages for berating referee Martin Atkinson.
Arsenal took their early lead when Ozil killed Kieran Gibbs' lofted ball down the left in an instant and just as swiftly rolled a cross into the path of the Frenchman, who swept it past goalkeeper Keiren Westwood with the minimum of fuss.
Sunderland were very nearly level within three minutes when an Arsenal defence deprived of the services of Per Mertesacker through illness allowed Modibo Diakite to head against the crossbar from a corner, and Altidore forced Wojciech Szczesny to beat away a long-range effort.
However, that was as good as it got for the hosts, who had Westwood to thank for keeping them in the game. Twice in the space of 13 minutes, Ozil sliced them open to put Walcott in on goal, but the England international could not find the right finish.
Walcott might have opened his account two minutes before the break when he met Giroud's cross with a diving header, but he was unable to hit the target.
Di Canio looked to provide his defence with greater protection when he replaced David Vaughan with Gardner at the break, and the newcomer was to make an unexpectedly early impact.
Koscielny was adjudged to have tripped skipper-for-the-day Johnson in the box after the visitors failed to deal fully with a corner and Gardner, who had been on pitch for less than four minutes, sent Szczesny the wrong way from the spot to level.
Suddenly, Sunderland looked a completely different proposition as they took the game to the visitors, Steven Fletcher seeing a 59th-minute effort ruled out for offside and Gardner hitting the post with a free-kick two minutes later.
But Arsenal managed to regroup and regained the lead with 23 minutes remaining when Jenkinson crossed from the right for Ramsey to smash a fine volley past Westwood.
Sunderland thought they had drawn level for a second time moments later when Altidore showed good strength to hold off Bacary Sagna and Szczesny could not prevent his shot from trickling over the line, but referee Atkinson had already blown his whistle for a foul by the Arsenal defender outside the area and play was called back for a free-kick.
Ramsey rubbed salt into the wounds six minutes later when he made the most of the link-up between Ozil and Giroud to cement the win.
Di Canio departed prematurely after making his feelings plain and drawing the attention of referee Atkinson, but he was at least able to console himself with a much-improved second-half showing.