Martin Odegaard had not taken a penalty for Arsenal before netting against Palace
Eddie Nketiah was brought down by Sam Johnstone after Gabriel Martinelli's quickly taken free-kick, with Odegaard sending the goalkeeper the wrong way from the spot.
At that stage Mikel Arteta's side looked comfortable, but they were left hanging on after left-back Takehiro Tomiyasu was shown a red card for two bookings.
The first yellow was for taking too long over a throw-in and the second was harshly awarded for a minor foul on Jordan Ayew in midfield.
Palace put on late pressure, with Tyrick Mitchell drilling over in injury time, but the Gunners, who hit the post through Nketiah in the first half, deserved their victory.
Arsenal join Brighton and Manchester City on six points from two games, while Palace failed to build on their opening-day win at Sheffield United.
Arsenal have a bit of a problem at full-back. Summer signing Jurrien Timber suffered a knee injury last week that could keep him out for most of the season and Oleksandr Zinchenko has just returned from a calf injury and started on the bench.
Tomiyasu, who is a right-back by trade, started on the left and Thomas Partey, a midfielder, played at right-back when Palace had the ball, moving into his usual position when his side had possession.
Japan international Tomiyasu will miss the next game now after his controversial red card. He was shown a yellow card for taking too long over a throw-in - although it was Kai Havertz who had wasted most of the time before leaving the ball for his team-mate.
And then he was dismissed for a nothing foul on Ayew about 50 yards away from any dangerous area.
That was Arsenal's first Premier League red card since 2021-22.
The Gunners were the better team for most of the game, and almost led when Nketiah poked a shot off the post and then chipped an effort onto the roof of the net.
Roy Hodgson's Palace defended stoutly for 99% of the game but their one blip effectively cost them the game.
They were taking their time to organise their defensive shape from an Arsenal free-kick when Martinelli caught them out with a ball to Nketiah, who was felled by Johnstone. Odegaard, whose last penalty was for Vitesse in 2019, made no mistake from the spot.
Declan Rice had another solid game in the centre of midfield following his £100m move from West Ham, but Havertz is still trying to find his proper role in the team.
Six of Palace's 14 chances came in the final five minutes with Odsonne Edouard having an effort saved and Mitchell lashing over after Aaron Ramsdale parried the ball.
-- Courtesy of BBC Sport