Rogers opened the scoring early for Xabi Alonso's side in the 2-1 defeat, netting for the second time since his British-record £117 million move from Aston Villa this summer, but goals from Kai Havertz and Martin Ødegaard turned the game around as Arsenal claimed a 12th successive game unbeaten against Chelsea across all competitions.
"We are disappointed. We came here with a job to try to win, and we didn't manage that, so of course the players and the boys are disappointed. But it's a long season and it's a tough place to come," Rogers said.
"I think we gave a good account of ourselves. We played well in large parts of the game, created chances and defended well when we had to. Of course, it's tough, and they're in a really good moment right now and feeling really good.
"There is still stuff for us to work on. We're still new, with a new manager, trying to combine our ideas and get better and better. But we will. We know we will. We know we've got the quality in the room. It's just about working hard on the training field and putting things right."
Rogers was jeered by Arsenal supporters after being substituted in the second half following a niggly exchange with Declan Rice that earned him a yellow card, with home fans chanting "You should have signed for a big club" as he made his way to the away dugout — a reference to the Gunners' interest in signing him from Villa before Chelsea smashed their transfer record to complete the deal. He responded by smiling and waving to the crowd rather than reacting.
Despite the mockery, Rogers gave no indication of any regret over his choice of club.
"It's been really good and really positive," he said. "You see time and time again that managers come in and struggle at the start when they are adapting, but we have done really well to take the manager's ideas on board. We have used our quality and the players we have in the changing room. We have done well to win the games we have and with the way we have played.
"Of course, there are areas to improve because it is really early days. But seeing the positive signs so early and being encouraged constantly, every day on the training pitch and in games, there is only one way we are going — and that is forward. That is all you can ask as a player when there has been a change in the building and you are trying to achieve something."
Rogers has scored twice and set up another since his move, having also found the net on his debut against Fulham a fortnight earlier. Chelsea, who remain without European football this season after finishing 10th last term, return to action on Wednesday when they host Leeds in the third round of the Carabao Cup.