The Gunners struggled to cope with Riyad Mahrez in the opening stages and the winger's vision led to the game's only goal as he picked out Phil Foden, whose drive was parried straight to Sterling by goalkeeper Bernd Leno and the England man calmly converted.
Beyond that Mikel Arteta's side produced a largely-assured performance and Bukayo Saka came close to a leveller, but Arsenal struggled to penetrate City's defensive line and the hosts held on comfortably.
For now at least, Arteta's men remain just a point behind second, while Saka continued to show glimpses of why he's such an exciting prospect. If nothing else, the visitors worked tirelessly after the break and on another day might have snatched a point but for the impressive Ederson in the City goal.
Defensive duties are a daunting prospect for any side arriving at the Etihad and Rob Holding's injury (sustained in the warm up) didn't help. David Luiz came in to replace him and the Gunners weren't awful, but they never really looked like moving through the gears as they struggled for clear-cut chances
Manager rating (out of 10)
6 -- It didn't really happen for Arteta's side here, but Arsenal held their shape well in a disciplined display after some early pressure. Despite that, the Gunners were perhaps too conservative and the manager barely threw a teaspoon at City, let alone the kitchen sink.
Player ratings (1-10; 10 = best, players introduced after 70 minutes get no rating)
GK Bernd Leno, 6 -- Showed strong hands to beat away Foden's effort, but he pushed the ball straight to the grateful Sterling. The German made a series of smart saves thereafter, first denying Mahrez and later Foden.
DF David Luiz, 6 - It was a matter of inches at both ends for the Brazilian, who poked Joao Cancelo's low cross just wide of his own goal before his Kevin De Bruyne-esque pass almost picked out Pierre-Emerick Aubameyang.
DF Gabriel, 7 -- He showed plenty of authority early on in a typical centre-back's display, repelling Bernardo Silva's cross before heading away the resulting corner, but he was punished for gifting City too much room as Sterling pounced.
DF Kieran Tierney 6 -- Never shy to charge forward down the left and provide attacking support, but he struggled against the pace and flair of Mahrez who was a constant thorn in Arsenal's side in the first-half. In fairness, he was more assured after the break, although he was schooled by Cancelo.
MF Hector Bellerin, 5 -- A steady start to the game, but Bellerin didn't cover himself in glory as he showed Foden the inside far too willingly in the build-up to Sterling's opener and the latter had him seated in the second half as City searched for the killer goal.
MF Dani Ceballos, 6 -- Had City players back-peddling at speed when he galloped forward before slipping the ball wide for Aubameyang and he produced some calculated passing in the middle. That said, he was completely taken out of the game by Mahrez's superb vision, but he produced a nice nutmeg.
MF Granit Xhaka, 6 -- Expertly read the danger to hurl himself in front of Sterling's ambitious early drive and applied himself well in the middle, releasing Aubameyang and later Saka on the flank as he did his best to turn the screw in Arsenal's favour.
MF Bukayo Saka, 7 -- Twice came close to the leveller and was twice thwarted by Ederson, but he was Arsenal's brightest spark and showed glimpses of his potential, including a mouthwatering one-two with Aubameyang to beat the offside trap.
MF Nicolas Pepe, 6 -- Pressed well to interrupt City's attempts to play out from the back but he struggled to source opportunities in front of goal. When a rare opportunity did fall his way he could only head Aubameyang's cross straight at Ederson and received a knock for his troubles.
FW Willian, 5 -- The former Chelsea man has impressed since his move across London, but a series of disappointing set-piece deliveries let City off the hook after some promising play and he found himself dragged into deep areas too often.
FW Pierre-Emerick Aubameyang, 6 -- Always willing, but without reward. He showed electric pace to burst away down the left and earn a corner, and applied himself again to quickly close down Ederson who was fortunate to see his attempted clearance deflect safely out for a throw.
MF Thomas Partey, NR -- Arsenal's £45m new arrival replaced made his first appearance as an Arsenal player but had minimal impact, although he picked up a late booking.
FW Alexandre Lacazette, 6 -- Replaced Willian for the last 20 minutes or so but, like the Brazilian, he found himself being dragged into deep areas where he struggled to cause problems.
FW Eddie Nketiah, NR -- Brought on for Pepe late on as Arteta looked to sharpen his attacking spear, but he struggled to get involved.