With John Stones on the bench for a second successive game, the central pairing of Aleksandar Kolarov and Nicolas Otamendi were caught cold as Alexis Sanchez split them with a well-paced throughball.
Walcott gratefully accepted the invitation and finished coolly, giving Claudio Bravo no chance.
City tried to respond quickly but another aspect of Pep Guardiola's selection was soon being questioned as Sterling spurned a good chance.
Sterling had started alone up front but he headed wide after being picked out superbly by De Bruyne.
Yaya Toure also headed at Petr Cech from a De Bruyne corner and the Arsenal keeper then had to react quickly after a Gabriel clearance rebounded in his direction off Hector Bellerin.
Despite these openings, City were generally sluggish.
They were almost undone again when Walcott planted a free header wide and Bellerin fired another effort over.
But Sane showed his talent in flashes and some nifty footwork saw him beat Nacho Monreal reach the byline but Gabriel cleared the danger.
He then made his next opportunity count to some effect two minutes into the second half. Replays suggested the German may have been marginally offside but, racing forward in a central position, he took David Silva's lofted ball in his stride with a clever turn. He beat Cech with a first-time finish.
The goal enlivened City and within moments they and Sane could have had another as De Bruyne played the 20-year-old in but this time Cech saved.
De Bruyne also let fly with a vicious low shot from distance that took a deflection off Granit Xhaka but Cech palmed away.
City continued to crank up the pressure and they took the lead after De Bruyne found Sterling with a brilliant crossfield pass from just inside his own half. Sterling still had plenty to do but he cut inside Monreal and rifled in a powerful low shot inside Cech's near post.
The hosts went close to another as De Bruyne hit the post at the end of another lightning attack.
Arsenal substitute Alex Oxlade-Chamberlain was taken off in the 78th minute, just 13 minutes after coming on. It was unclear if the England international was injured as he headed straight down the tunnel.
Arsenal were beaten and failed to threaten again as time escaped them.