City took a step towards the last 16 by keeping a clean sheet in Greece.
It was a professional performance given that Aris have never be been beaten in 40 years of European competition at the Kleanthis Vikelidis Stadium.
It could have been even better but former Wolfsburg striker Dzeko was out of luck in the second half.
Dzeko was one of four changes to the side that lost 2-1 to Manchester United in the derby on Saturday. Jerome Boateng, Kolo Toure and Shaun-Wright Phillips were all recalled to the side.
The atmosphere in the stadium was electric as the game got under way with fireworks lighting up the night sky just before kick-off.
It was easy to see why some many teams have failed to cope with surroundings over the years.
City had the majority of possession in the first half but had nothing to show for their efforts.
They showed their intentions early on and David Silva cracked in a shot from 20 yards after nine minutes that was turned away at the post by goalkeeper Michalis Sifakis.
Then Carlos Tevez tried his luck from distance five minutes later but his effort finished wide of the target.
Tevez was playing a deeper role than normal with former Wolfsburg striker Dzeko on his own up front.
City stepped up the pace and Dzeko almost opened the scoring in the 16th minute.
He got on the end of a driven cross from Aleksandar Kolarov but saw his close range shot gathered by the goalkeeper at the second attempt.
City were keeping possession well and frustrating an Aris side who came into the match on the back of a home defeat by Olympiacos.
City launched a swift counter-attack in the 28th minute but Dzeko only succeeded in drilling his shot into the body of Sifakis.
Dzeko was finding plenty of room and four minutes later latched on to a ball from Gareth Barry but his finish was wayward.
Aris were finding it a struggle to get going but won a free-kick after goalkeeper Joe Hart handled outside the area.
Referee Alberto Mallenco allowed play to continue in the 39th minute but after consulting with his assistant brought Hart to book.
That appeased Aris, whose players surrounded the assistant after the ball ran out of play.
Neto took the free-kick but his effort went the wrong side of the post.
Wright-Phillips was becoming more involved and cut inside in the 57th minute only to send his effort into the body of Sifakis.
Back came Aris and Neto sent a free-kick over the wall but Hart was able to take it cleanly.
The home side were growing in confidence and Juan Toja sprinted forward in the 63rd minute and fired in a shot that Hart was equal to.
Dzeko again came close to giving City the lead a minute later following a cross from Barry. Dzeko got in a towering header but Sifakis somehow managed to tip it over the top of the bar.
Play then switched back down the other end and Neto was denied by Hart as the game suddenly opened up and Barry did well to block a shot from Michel.
City now knew they were in a game as the crowd cranked up the atmosphere even further.
Mario Balotelli was then handed his first outing in six weeks following a knee injury when he replaced Wright-Phillips late on.
Even though City failed to make the breakthrough they will be expected to complete the job in the second leg.