Kai Havertz opened the scoring at Estadi Montilivi after 16 minutes, finishing calmly when played through by 16-year-old Dowman, before deadline-day signing Tzolis doubled the lead on his debut in the 30th minute, cutting inside from the left and firing in off a deflection following a Riccardo Calafiori pass.
Girona pulled one back through captain Arnau Martinez's header early in the second half, but Arsenal responded within three minutes when Dowman drove home from outside the box, and Gabriel Jesus added a fourth soon after, pouncing on a defensive mix-up to seal a 4-1 win.
Speaking afterwards, Arteta was full of praise for both Dowman and Tzolis. On Dowman, who followed up a breakout pre-season last year with another eye-catching display, he said: "That's what we want. When you play for Arsenal, the expectations should be high and should be expecting really good things. We know what he's capable of doing. We want to be more consistent in terms of the performances and the actions that he can deliver. But I think today he had a really good game."
On Tzolis, making his first appearance since his £34m move from Club Brugge, Arteta added: "I'm very pleased with him. First of all, the way he has adapted to the club, to the team. The connection he's already generating with the boys. And I think you're going to love the player.
"A player who's very direct. He's got the goal between the eyes. I think really good linking up. He has a lot of similarities with Leo in a lot of the things that he can do and the spaces he can occupy. And then the work rate that he has. He's a boy that comes with so much hunger as well. He wants to prove himself, it's a dream for him. And we are here to help him."
Arteta also discussed Ethan Nwaneri's return to midfield after a difficult loan spell at Marseille last season.
"On the pitch, he didn't [go as well as we wanted] because a lot happened in Marseille. But when you just stand in front of him, you sense the maturity. And it looks like a different man," Arteta said. "So, yeah, the experience on paper, it was an idea. But I think he went through something that is going to help him a lot in his career."
Asked more generally about the transfer market, with Bruno Guimaraes closing in on a move to the club, Arteta confirmed further signings remain possible before the season begins.
"We are really active, trying to improve and evolve the team. That's clear. We expect to have movements in the next few weeks, obviously, because we want to get better like anybody else. And you can just see it. The transfer market and our opponents, what we are doing, we won't sit still. And we are very ambitious in what we want to do," he said, explaining why Arsenal continue to strengthen despite being reigning champions and Champions League finalists:
"Because the margins are very small. And because we want to get better and the level is going to increase, we need to increase the competition internally. We need to make sure we identify the things that we don't have in the team, to have bigger margins. And that's the way we have to think."
On the club's academy contingent more broadly, having fielded several young players in the win, Arteta said he was using the pre-season fixtures to assess who might be ready to push for first-team involvement this season: "That's a great opportunity to see them live. They've been almost two weeks now with us. It's great to see how they react to actually a competitive match. I'm really happy with what I've seen."
Asked whether being champions for the first time changed how he approached pre-season, Arteta was clear it had only sharpened his focus: "I think we have even more responsibility and more ambition than before. We know what is ahead. We know the qualities and the capacity that we have to compete with any other team. But we need to prove that we belong to this level."