The England team-mates clashed after Rice tripped Rogers during the match at the Emirates, with the Chelsea winger reacting angrily before Arsenal supporters taunted him over his decision to join Chelsea.
Rogers insists there is no lingering tension between the pair.
“It’s just competitors competing and wanting to win,” Rogers said.
“London derby, me getting absolutely dog’s abuse by the fans doesn’t help! All emotions come into one at that moment, but Dec and I are fine.
“We spoke about it after the game. We’re completely fine. It’s just one of those things in football, we want to win.”
The incident subsequently became the subject of viral memes and AI-generated videos, with Rogers also receiving fake quotes attributed to him.
“It’s quite funny being sent them. AI’s going to the next level now at the moment. Anything you do now can end up in anything,” he said.
Rogers has made a positive start to life at Chelsea after choosing to leave Aston Villa during the summer.
The £117 million signing said the opportunity to join one of the Premier League’s biggest clubs was difficult to resist.
“When they come in and there’s an opportunity for me to go there, it’s mind-boggling to think that I can be part of that history of that club,” Rogers said.
“I’m so excited by the prospect of doing something special at that club.”
The 23-year-old also accepts the pressure that comes with his price tag but believes Chelsea provides the right environment for him to continue developing.
“I just want to give positive performances for my country and to achieve something special with my country,” he said.
“And then the calibre of players I’m playing with and training with every day, that’s taken a whole new level and it’s made me have to improve and get even better.”