The youngster, who was awarded the Man of the Match prize, set up the first goal of the evening, beating his man and finding Bernardo Silva expertly to tap in from close-range.
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Gabriel Jesus and Ilkay Gundogan extended the lead, before the hosts pulled one goal back through Harry McKirdy.
However, Palmer settled the contest with a sumptuous finish with eight minutes remaining to cap off a wonderful display from the Englishman. He said:
"We started well. We knew it’s a tough place to come so I’m delighted for the win.
"I enjoyed out there. I was told to stay high and wide and I tried to create things and luckily it paid off.
"I’m delighted, I’ve just got to take it game by game and day by day. There’s always that bit of pressure being young coming into this team. I've just got to go out there and enjoy it."
The 19-year-old is eager to remain grounded despite the performance and is keen to make the most of the minutes that he's rewarded with by manager Pep Guardiola.
"He [Guardiola] has been in touch with Rodolfo and he’s then passed the information onto us.
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"Hopefully he enjoyed my performance and I just have to show what I can do day in and day out.
"I have to just keep going, get more goals and more assists and help the team out."