Gunners find themselves three points off the top
An early red card for Per Mertesacker proved crucial as Arsenal slipped up at home to Chelsea.
A 1-0 defeat for the Gunners sees them slip to third just two points above local rivals Tottenham and three below Leicester City.
Arsenal suffered a very early setback when Costa latched onto a ball over the top of their defence. A last ditch (and last man) challenge from Mertesacker saw him sent off just 18 minutes in as he felled the Brazilian-born Spain international.
Chelsea, who started with a solid midfield base of Nemanja Matic and John Obi Mikel, capitalised soon after as Costa proved decisive again.
Five minutes after the Mertesacker red, Costa turned a Branislav Ivanovic cross past Petr Cech in the Arsenal goal.
Despite the disadvantage on the scoresheet and in numbers on the field, Arsenal dug in and had their chances.
Just before half-time, Mathieu Flamini fluffed an opportunity six yards out when he acrobatically prodded the ball over the bar when a header might have been an easier way to hit the target.
Chelsea now rise to 13th in the Barclays Premier League table as their unbeaten run under Guus Hiddink rises to nine games, while Arsenal stay third on goal difference after Manchester City earned a draw at West Ham yesterday.
Next up for Arsenal in the league is a home match against Southampton on Tuesday, February 2nd, while Chelsea visit Watford the following night.