Eight minutes into the half, Chelsea were ahead as Marcus Alonso broke the down the left flank before teeing up Pedro, who slotted into the bottom corner.
The Gunners should have leveled matters in the 18th minute when Hector Bellerin teed up Pierre-Emerick Aubameyang, who fired over the crossbar from 10-yards out.
Arsenal were made to pay for that miss as Alvaro Morata doubled Chelsea's lead a minute later with left-footed strike after latching onto a long-pass.
The Gunners did a pull back 10 minutes before the break with Henrikh Mkhitaryan firing into the bottom corner from the edge of the box.
Two minutes before the break, Alex Iwobi brought the visitors level with a clinical finish from 12-yards out after being picked out by Mkhitaryan.
The second half continued in same fashion as the first, with the Blues dominating possession as they looked to restore their lead.
On the hour mark, Chelsea introduced Eden Hazard and Mateo Kovacic in place of Ross Barkley and Willian.
Hazard made the perfect impact as he created Chelsea's third goal by picking out Alonso in the box, and the Spaniard calmly slotted into the far corner.
Aaron Ramsey almost brought the visitors level in the closing stages, but his long range effort was tipped over the crossbar Kepa Arrizabalaga as the Blues hung on for victory.