The Gunners defenders had to be pulled apart by team-mates and staff at the full-time whistle, with Arteta explaining they were unhappy at conceding late.
Arteta admitted things got “heated” between the pair, but he was pleased as it shows they are “demanding more from each other”.
The win over Forest was a vital one for Arsenal as it moved them up to second in the Premier League and two points behind leaders Liverpool.
“I love it,” said Arteta, when asked what happened with Zinchenko and White. “They are demanding more from each other.
“They are not happy with the way they conceded and they are just trying to resolve it. It got a bit heated. But that means that it’s enough. Playing the way we played the result has to be bigger and the clean sheet has to be there.”
Gabriel Jesus was Arsenal’s hero as they claimed a 2-1 win. The Brazilian scored the opening goal of the game and then set up Bukayo Saka for the second.
Arteta was full of praise for Jesus’ performance and revealed how he was a major doubt to play in the game.
“Gabi started to win the game two days ago,” said Arteta. “He had an issue with his knee and everyone was trying to protect him and saying don’t go outside.
“But he was saying (matchday) -2, (matchday) -1, I want to be there I want to help the team to win the game. When you have that mentality, good things are going to happen. I’m really pleased with him.
“He got hit big time in the last game. His knee reacted and it’s the knee he had (surgery on) before.
“He was super positive, he’s feeling good and he was so sharp in training. I’m not surprised with the way he played tonight.
“He could have scored more today when he hit the post and had the header. His general activity, when he had to link play and generate chaos he was there. He was really good.”