The two former Arsenal players were working on the match which the Gunners won 2-1, thanks to a brace from Pierre-Emerick Aubameyang.
Cole played for both Arsenal and Chelsea, though clearly wanted Frank Lampard’s side to lift the cup at Wembley.
And Wright filmed himself celebrating with the video posted on social media.
Wright said to camera: “I’m really happy, look how happy I am.”
Cole is then seen looking over Wright’s shoulder with the former Gunners striker adding: “Unlucky Ash”, while laughing away.
Wright then spots David Luiz on the pitch and screams: “David! Oiiiii! I love you. We did it Arsenal.”
Arsenal went behind to Christian Pulisic’s strike before Aubameyang levelled the scores from the penalty spot after the Gabon forward was fouled by Cesar Azpilicueta.
Aubameyang then scored the winning goal with a deft chip over Willy Caballero in the second half.
Gunners boss Mikel Arteta said: “It was a difficult start when we conceded so early. If I know something about these players, is that we weren't going to give up.
"We played probably the best 30 minutes after since I arrived. I am so proud to represent these players and this club.
"It is a double reward for us. It is really important for this club in every way to be in Europe. We all did it so I am so happy.
"The players were completely knackered and they made a massive effort through the coronavirus to keep fit so I am so happy for them."
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While Arsenal defender Rob Holding added: “It took us a while to get used to how they were playing but once we got settled we grew into the first half and then Aubameyang seems to do it on the break every game. He is a our talisman and has done it all season.
"14 FA Cups is crazy. It's my second in four years! I saw on Twitter all the Arsenal fans setting up so as long as they enjoyed it…”