Though regular taker Bukayo Saka was on the pitch, it was Odegaard who stepped up to convert the spot-kick that gave the Gunners their 1-0 win on Monday night.
"I had no idea Martin was going to take it," Arteta said. "It's their decision and then it's the right decision because they feel it, and if they feel it they have to do it regardless what I think.
"I didn't know. I expected Saka to take it and then he passed it to Martin and they felt it like this in that moment, and that's right because somebody has to take it."
The decision by Saka was appreciated by his captain.
Odegaard said: "We've been taking [penalties] in pre-season a lot and I've scored all mine, so I just asked him, and he gave it to me so I have to thank him for that one."
What pleased Arteta most was his side's resilience as they held on to their slender lead.
Arsenal had conceded in the 82nd minute against Nottingham Forest the previous weekend, turning a comfortable 2-0 lead into a nervy 2-1 victory.
The Gunners' resolve was then tested against Palace as Takehiro Tomiyasu's dismissal left them defending their 1-0 advantage with 10 men for half an hour.
"I’m so happy to win, always very happy to win, but very happy in the way we won with 11 players and 10 players, and the way the team adapted, and how much they wanted the win," Arteta told the club's official website.
"The contributions from the players on the bench and the way they changed the game straight away when they came on... It doesn’t matter the circumstances, it doesn’t matter the decisions, we keep going. We want to win and find a way to win it.
"They absolutely did the job, they didn’t concede anything, apart from one cross - I think it was a wide free-kick and they had a volley, but after that nothing. And we had some good moments as well when we changed the momentum of the game, so I’m really happy."