The captain stroked home a spot-kick on 53 minutes after Eddie Nketiah was brought down in the box by Sam Johnstone, before Takehiro Tomiyasu was dismissed for two yellow cards - our first dismissal in the Premier League since May 2022.
However Mikel Arteta’s team clung onto the points thanks to a solid rearguard display from that point onwards to kick off the season with six points from six.
Both sides had a number of decent chances in the first half-hour with the best of them falling to Arsenal's Eddie Nketiah, who hit the post in the 29th minute after some good pressing by the midfield saw the Gunners win the ball high up the pitch.
The striker was in the thick of things again early in the second half as he was felled in the box by keeper Sam Johnstone when running onto a quick free kick by Gabriel Martinelli.
Referee David Coote pointed to the spot and Odegaard sent Johnstone the wrong way, rolling the ball in at the foot of the keeper's right-hand post.
Arsenal looked set to cruise to victory but Tomiyasu was shown his first yellow card on the hour mark for time-wasting at a throw-in and was sent off in the 67th minute when he collected his second booking for a foul on Jordan Ayew.
Arsenal, meanwhile, battled to a 2-1 home victory over Nottingham Forest, with Mikel Arteta’s side hoping to go one better this term after finishing runners-up to Man City in May.
The Gunners have added the likes of Declan Rice and Kai Havertz to their squad over the summer, while Palace lost key winger Wilf Zaha - but have brought in Matheus Franca from Flamengo in their big-money addition so far.
Martin Odegaard took and scored his first penalty in a league game since doing so for Vitesse against De Graafschap in the Eredivise in May 2019. Indeed, since the start of last season, he’s just the third player to score at least 10 Premier League goals away from home after Erling Haaland and Harry Kane (both 16).
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