Chelsea had one promising attack after five minutes when new signing Raheem Sterling played Timo Werner into space, but French defender William Saliba snuffed out the danger with a solid covering tackle.
Arsenal should have opened the scoring moments later, when Gabriel Martinelli burst clear on the counter.
The Brazilian passed to Bukayo Saka but the English international's shot was saved by Chelsea goalkeeper Edouard Mendy.
The Arsenal pressure finally told in the 15th minute with a goal that came after Chelsea's Nathaniel Chalobah surrendered possession with a misplaced pass that went straight to Granit Xhaka.
The Swiss international played a superb first time ball into Jesus who coolly chipped over the advancing Mendy to make it 1-0.
Martin Odegaard made it 2-0 to the Gunners in the 36th minute with another well-worked move.
Martinelli's pass found the Norwegian playmaker in space inside the Chelsea area, and the attacking midfielder turned smoothly and tucked a low finish past Mendy.
Chelsea's best chance of a lacklustre half came in stoppage time when Mason Mount curled a shot from the edge of the area which beat Aaron Ramsdale but crashed back off a post.
Arsenal extended their lead midway through the second half when Saka buried a rebound from close range after Mendy parried Xhaka's long-range strike.