Despite a frustrating first half that saw Eddie Nketiah denied by the woodwork, Arsenal proved their grit in the second half when they went down to 10 men to take home three points from their travel to Selhurst Park. Odegaard's second half penalty was the only thing that separated the two teams in the end.
Odegaard's second-half penalty at Selhurst Park put Arsenal on course for their second successive win. But Mikel Arteta's side had to dig deep to preserve the points after Japan defender Tomiyasu was controversially sent off for a second yellow midway through the second half.
Tomiyasu had been booked for taking too long over a throw-in before his dismissal for making minimal contact on Jordan Ayew. Arteta will have been delighted with the way Arsenal refused to surrender their lead as they moved up to third place, behind leaders Brighton and second-placed Manchester City on goal difference.
Expected to mount a sustained title challenge following last season's late collapse that gifted the trophy to City, the north Londoners look up for the fight after building on their opening weekend win against Nottingham Forest.
While they haven't played their most fluent football yet, this could prove a significant building block in their bid for a first title since 2004. Arsenal had the best away record in the top flight last season and registered a division-best 10 clean sheets on their travels.
Arteta's men looked like road warriors again as they repelled Palace's late flurry of pressure. With new goalkeeper David Raya among the Arsenal substitutes after his loan move from Brentford, the spotlight was on Aaron Ramsdale as the England international fights to hold onto his place.
Ramsdale rose to the challenge with a confident display from the moment he turned away Eberechi Eze's long-range strike in the early stages.