The 117 million pounds British-record signing from Aston Villa marked his competitive debut by scoring Chelsea's second goal at Craven Cottage, restoring their lead after Josh King had cancelled out Joao Pedro's early opener. Cole Palmer then added a third after half-time as Xabi Alonso began his Chelsea reign with victory.
Rogers was particularly effective alongside Palmer and Pedro, with the attacking trio producing an encouraging display despite playing together competitively for the first time.
"It was a great game to play in. Obviously to score and help the team win, that's everything that I wanted from a debut," Rogers said.
"The quality of our play was excellent and it was so enjoyable out there. Fulham is a difficult place to come to, and they played really well, but we had that fight, that grit and I think we were the better team on the night."
The 24-year-old was substituted with around 10 minutes remaining after 82 minutes of action. Having represented England at the World Cup just five weeks earlier, Rogers admitted he is still working towards full fitness.
"I didn't feel that sharp towards the end, I was a bit tired and that's why I came off," he said. "I've not had much of a break due to the World Cup, so I wanted to keep myself fresh.
"Coming to a new team, you don't want to come in sluggish and off the pace, so I tried to keep myself in the best shape as possible and I've tried to get up to speed as quickly as I can.
"I still think there are more gears to go, and I've got to try and find them as quickly as possible because the games are coming thick and fast."
Chelsea return to action against Luton Town in the League Cup on Thursday, when Rogers could get another opportunity to build on an impressive first outing.