Dowamn was dismissed for violent conduct in the closing stages of the match with the scores tied at 1-1.
The Gunners went on to lose on penalties against their EFL League Two hosts on a historic night for the Lancashire outfit as they set up a Carabao Cup quarter final date with Everton.
The post-match conversation was dominated by Dowman's rush of blood to the head, with the 16-year-old now banned for three domestic games, after Arsenal's appeal was rejected by the FA.
Arteta admitted to holding Dowman to account at full-time and he reiterated that comment ahead of a Premier League trip to Liverpool this weekend.
"This is the environment, this is the reality. We are not in Disneyland, we are in the Premier League.
"We look after him so much, but at the same time we're going to be demanding, because he needs to earn the right to play, just like everybody else, because the other boys, what are they going to say to me?
"You need to be fair.. and when we don't win, I'm p***** off.
"My reaction came from a moment of heat and extreme frustration, because I know the consequences of that, especially for Max.
"It's something we discussed just before we went out because it's not the first time it's happened, it happened another time with the England Under-19s.
"And it'll continue to happen because of the way he plays and the way he attracts opponents and that's it."