The win meant Arteta lifted his first trophy as a manager and Arsenal's club website have released a 'Bench Cam', detailing the Spaniard's emotions throughout the game.
When Christian Pulisic scored the opening goal and gave Chelsea the lead, Arteta immediately responded by barking instructions at his players.
He was soon celebrating himself, though, when Pierre-Emerick Aubameyang's penalty drew Arsenal level.
Just a few minutes later, Chelsea captain Cesar Azpilicueta was unfortunately forced off with a muscle problem and Arteta looked to comfort the defender, who was in tears.
In the second half, Aubameyang's magnificent second goal of the game saw Arsenal take the lead and Arteta's celebrations were much more animated as he leapt into the air on the touchline.
That goal ultimately proved to be the difference and Arteta celebrated with his staff when the final whistle was blown before sprinting onto the field to join his players, giving two-goal hero Aubameyang a big hug.
Speaking after the win, Arteta said: "It's a good start. Thank you so much for all the time being so supportive, from the reception I got on the first day.
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"When I walked through this incredible door I completely understood that the needs of this football club are so high. I know the demands and expectations.
"It's a trophy, it's absolutely for them to take, to enjoy to be proud of the club they support and the players that they always cheer for. Hopefully they can enjoy it together, maintain the distance please and enjoy because it's all yours.
"I knew the challenges I was facing the day I made a decision to come here and the expectation that we're always going to have because the history and the legacy of this football club is incredible.
"But here is the first one. Let's enjoy it today. We know there's a long way to go to improve on many other things but it's something that I am very proud of, how we have changed a lot of things since we joined.
"The players have been fantastic, the staff, thank you so much as well to the people upstairs who have given me the opportunity to represent and to manage this incredible football club.
"Again thanks to the people for their support and their belief."