The Gunners eased to a 5-0 victory to move back to the top of the Premier League and leave the visitors rooted to the foot of the table.
England forward Saka seemed to take a knock in the first minute but still teed up Gabriel Martinelli to break the deadlock.
He later limped off to be replaced by Nelson, who scored twice in three minutes before assisting Thomas Partey as captain Martin Odegaard wrapped up the win.
Nelson had not scored a top-flight goal in 837 days and this was the 22-year-old’s first league appearance since the opening day of last season, but Arteta was delighted with his impact.
“I think not only me but all his team-mates, all the staff are so happy for him because he is a kid that is really changing, he’s really evolving, he’s really maturing,” Arteta said of Nelson.
“He’s showing every day how much he wants it and today he had the opportunity and he did extremely well because it helped us win the game.”
On what had changed in Nelson’s game, Arteta added: “How mature he is, how focused he is in his profession.
“The way he talks, the way he reflects, the way he respects decisions, the way he is challenging everybody every single day.
“He is a joy of a kid. We all want him to be successful and do well. What he has done today for the team is something he really merits.”
Saka played on until the 27th minute before making way for Nelson. While the sight would have been worrying for Arsenal and England fans alike, Arteta does not believe it is a serious injury.
“Hopefully not,” Arteta replied when asked if Saka was a World Cup doubt.
“It was a bad kick. Right from the beginning he was limping, but I don’t see any further than that. Let’s see how he is in the next couple of days.
“He got kicked a few times and he wasn’t comfortable.”
Forest were largely outplayed as they failed to follow up last week’s shock win over Liverpool with victory on their first outing at the Emirates.
Despite not being in the game, boss Steve Cooper wants his players to learn from their thrashing.
“We don’t just have to take that on the chin – we have to deal with it,” he said.
“We fell way short today in terms of competing. Of course we could have played really well here and not go anything out of the game. It was always going to be a tough game.
“It ended up as painful viewing and painful for the players to play in.
“We’ve got to stand up now and challenge each other on how we fell well short today and do our best to put it right next week against Brentford.”