Gunners centre-back Luiz clipped Willian Jose’s heel in the box in a 2-1 defeat at Wolves, while Bednarek was adjudged to have fouled Anthony Martial in Saints’ loss at Manchester United.
An Arsenal statement said the club were “disappointed” with the appeal result.
Defender Bednarek will now be available for Saturday’s trip to Newcastle.
The Gunners were 1-0 ahead when Luiz was judged to have fouled Jose and, because there was no attempt to play the ball the defender was dismissed and Ruben Neves scored the resulting penalty.
Craig Pawson was the referee in charge – and following a video assistant referee check the decision remained, although Pawson was not asked to review the incident on the pitchside monitor.
Luiz, 33, will now miss Arsenal’s Premier League game at Aston Villa on Saturday.
Arsenal boss Mikel Arteta said earlier on Thursday that he was “very hopeful” the decision would be reversed.
I’m very hopeful. That’s why we put the appeal in. We went through all the images and we think the red card should be overruled. That’s why we appealed,
We know the law, for me it’s more about the decision and what happens in that instance.
He was on the side of the player that completely changes the angle, the purpose and the player’s ability to finish the action, but its still a similar action. I think they will look at it. said Arteter