The Blues eventually ran out comfortable winners at Stamford Bridge after the Belgian forward won and converted a first-half penalty and Cesc Fabregas inspired their second goal
Eden Hazard scored the opener from the penalty spot in the first half after Laurent Koscielny fouled him in the box, his cool spot kick sending Wojciech Szczesny in the wrong direction.
Playing against the Gunners for the first time since joining Chelsea this summer, Cesc Fabregas provided a delightful assist for the second goal as, with just over 10 minutes of normal time left, he lifted the ball over Arsenal's defence for Diego Costa to chase and finish with aplomb - the striker's ninth goal in seven league games this season.
Chelsea moved five points clear of nearest rivals Manchester City at the top of the table - the only downside of the game for them an injury to first-choice goalkeeper Thibaut Courtois, who was forced off after suffering a nasty blow to the head in an early collision with Alexis Sanchez.
The defeat saw Arsenal's unbeaten start to the Premier League campaign brought to an end, as Arsene Wenger's men fell nine points behind their London neighbours after just seven matches.
Both sides made one change following their exploits in the Champions League in midweek, as Cesar Azpilicueta returned at left-back for Chelsea in place of Filipe Luis and Jack Wilshere came in for Alex Oxlade-Chamberlain for the visitors.
After a 15-minute delay to kick-off, reportedly due to a number of flares being let off outside the ground, there was a moment of concern for Chelsea in the 10th minute as Courtois collided with Alexis. The Belgium international was able to get to his feet and resume action but he was clearly still shaken from the clash and had to be withdrawn in the 24th minute as Petr Cech entered the fray.
With little happening on the field, the opening stages were also notable for an argument on the touchline as Cahill's heavy challenge on Alexis provoked an irate response from Wenger, who pushed Jose Mourinho in the chest.
The Portuguese gestured for Wenger to return to his technical area before Atkinson called both managers over in an attempt to calm matters.
Finally, after 27 minutes, a moment of magic from Hazard led to Chelsea breaking the deadlock. The Belgian drifted beyond Santi Cazorla and Chambers, before being upended by Koscielny's trip inside the box and Hazard, who scored a penalty in the 6-0 rout last season, calmly sent Wojciech Szczesny the wrong way from the spot.
Wilshere's poor touch then allowed Cech to gather as Arsenal went in search of an immediate equaliser.
Chelsea almost doubled their lead shortly before the hour when Hazard's driven cross deflected off Mathieu Flamini and then the arm of Szczesny before Arsenal felt that they should have had a penalty when Wilshere's effort hit the arm of Fabregas, although it was unclear whether or not the incident had occurred inside the box.
Hazard ought to have netted his and Chelsea's second when he fired off target from close range 14 minutes from time but, soon afterwards, Costa made the game safe. He ran onto Fabregas's pass and lobbed a deft finish over the advancing Szczesny.
Danny Welbeck was perhaps lucky to only receive a yellow card for a two-footed lunge on Fabregas in stoppage time as Arsenal desperately chased the game but Chelsea comfortably held on to secure maximum points.