In a clip posted on Twitter by Gary Lineker, Wright leapt from his seat and yelled "Yes, come on!" after Pierre-Emerick Aubameyang scored his first-half penalty.
Wright was at Wembley for the final alongside Lineker, Alan Shearer and Ashley Cole as part of the BBC Match of the Day coverage and Lineker captioned Wright's celebrations: "He’s cheered up has @IanWright0 "
Chelsea took the lead in the final thanks to a goal from American forward Christian Pulisic but Aubameyang got Arsenal back into the game.
The Gunners captain was brought down in the area by Cesar Azpilicueta and confidently dispatched the spot kick, prompting Wright's celebrations.
Aubameyang then put Arsenal ahead with an excellent finish in the second-half, beating Kurt Zouma and dinking the ball over Willy Caballero in the Chelsea goal.
"Wow! What a finish! What a player!" Danny Murphy said of Aubameyang's second goal on BBC One.
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"Nicolas Pepe picks Pierre-Emerick Aubameyang out and he's onside. Look at that dink... He's a world class footballer. We have seen how important it is today for Arsenal to keep him. It's a beautiful, beautiful finish."
Newcastle United legend Alan Shearer added: "It's exceptional the way he gets in there, turns the defender and the finish is sublime. He is red hot! What a finish!"
"You can see why Arsenal want to keep Pierre-Emerick Aubameyang," former Chelsea striker Chris Sutton said on commentary for BBC Radio 5 Live.
"It is absolutely genius by Aubameyang. He drifts past Kurt Zouma and dinks the ball over Willy Caballero. It's one of the great goals - an absolute work of art. It's sad Wembley does not have any fans in it.
"Pierre-Emerick Aubameyang would cost zillions to replace. Absolute zillions."