Mikel Arteta’s side conceded their first goal of the season just two minutes into the game after Morgan Rogers thrashed home a lovely volley.
Arsenal didn’t worry, however, and eventually got a well-deserved equaliser through Kai Havertz before Odegaard, 27, bagged the winner in the second half.
The North London club now have three wins from their first three games of the Premier League season, sitting in second, behind Man City on goal difference.
Speaking to Sky Sports after the game, Odegaard was delighted his side secured a ‘massive’ win against their London rivals.
"It was a big game, massive game, tough opponent and we got there in the end so brilliant," he said.
"Our mentality is to keep fighting. We have shown that many times. We never want to concede early but the reaction is top.
“We got our reward in the end and the response was brilliant. We showed our quality with and without the ball and the result reflects that.
On his relationship with Kai Havertz: "He's a really smart player, he moves so well and takes a lot of attention from the defenders and it was a brilliant goal."