Palmer was central to Chelsea's 3-2 win, setting up Joao Pedro's rapid opener before scoring the team's third himself, and reflected afterwards on both the win against Fulham and his own comeback from an injury-hit campaign.
"It's important — away game first is always tough but we dug in and got the win," Palmer told Sky Sports.
Asked how ready he felt for the new season, he offered a typically dry response before turning serious.
"Did you not watch the game? I'm only joking, I'm ready you know. It was a difficult season with injuries and whatnot but through the summer and since I've been back I've worked extra hard. Obviously with the injuries I've had to try and prevent them and work extra hard day in, day out.
"I know the talent I've got so I can put in performances like that."
On his mindset heading into the campaign, Palmer struck an unusually reflective tone.
"Normally I'm not an emotional person, the people who know me know I'm not usually bothered about things in life, but over the summer I rested, I recharged and yeah I'm ready and I'm hungry."
The afternoon carried extra significance for Palmer as he lined up alongside Morgan Rogers, a close friend since their days coming through England's youth ranks together.
"When you grow up you want to play or do anything with your best friends, so playing football at the best level feels surreal," Palmer said.
Rogers, who marked his Chelsea debut with a goal, echoed those sentiments.
"It was really nice. It was something we've spoken about for a while, obviously, we've done it on an international level but to do it for the club week-in-week-out and day-in-day-out is an honour really. It's just the start," he told Sky Sports.
The forward also described the emotion of hearing Chelsea's supporters take to him so quickly.
"It's nice — you want to settle in as quickly as possible, integrate into the squad and with the fans and feel that affection in the first game and obviously to score is a really nice moment for me."
Rogers had little hesitation in singling out Palmer's performance as the difference-maker.
"Everyone knows how good he is. Obviously, he had a difficult season with injuries and stuff last year and you could see it in his eyes in pre-season and training that he's ready this year and ready to prove people wrong. You saw that with that performance — the best player on the pitch by a mile and that's what he does. It's good to have him on our side."
Both players also offered an early verdict on life under new head coach Xabi Alonso.
"There's still a long way to go and things we need to learn and get better and improve at but it's a positive start," Rogers said. "We've tried to implement how he wants to play but it's a process and it takes time with new players integrating as well. But you want to start well and have a good platform and we did that today."