However, the 16-year-old was sent off with five minutes remaining, as Arsenal finished the match with ten men.
Dowman reacted angrily to a heavy challenge from Harrison Neal and kicked out at the defender, even if the force was minimal.
Red card: Max Dowman
AFP/Getty
He was shown a straight red card and as he walked off the pitch and down to the tunnel, Arteta lent towards Dowman with his arms crossed and aimed some furious words in his direction.
Asked after the match what he had told Dowman, Arteta told BBC Radio 5 Live: "Something that we discussed already before the match."
That suggested Arteta had warned the teenager in the build-up to the fixture to expect rough treatment and not to react to it.
Despite his reaction on the touchline, Arteta was reluctant to discuss the incident leading to the red card.
"It's difficult to see on the cameras like that," Arteta said. "So I prefer to watch it back properly and then I can assess the situation."
Something that we discussed already before the match
Mikel Arteta
Even with Arteta significantly rotating his side, the likes of Kai Havertz, Gyokeres, Martin Zubimendi, Myles Lewis-Skelly and Jurrien Timber all started.
Martin Odegaard and Ben White were introduced at half-time, while Guimaraes, Ezri Konsa and Eberechi Eze all came on later in the half.
Still it was not enough for Arsenal to see off Fleetwood, but Arteta refused to criticise the mentality of his team.
"I cannot fault at all the attitude of the players," Arteta said.
"I haven't seen anyone not doing what is required at this level. But, obviously, we insisted so much, we have so much of the ball, so many chances, so many shots, and if you want the margins to be bigger than going to the penalties, obviously, you need to capitalize on that.
"And we didn't. And football is about that."