The winger came on for Erling Haaland, who had a quiet game and failed to test Aaron Ramsdale in the Arsenal net, but Palmer had no such problem.
And Manchester City fans were left with their jaws on the floor after the 21-year-old curled an incredible strike past the Gunners goalkeeper late into the contest.
One fan wrote: "Cole Palmer that is sensational!!!!"
Another posted: "Class attacking play from City, class goal from Palmer."
And a third said: "Golazo, Palmer!"
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This fan put: "Local lad and childhood Manchester City fan, delivering for the team he grew up supporting."
While this fan wrote: "Cole Palmer, that is majestic."
And speaking on talkSPORT, Stuart Pearce was left in awe.
Commentating on the match alongside Jim Proudfoot, he said: "Five minutes ago I suggested Palmer shout be aiming for that far corner, on that occasion he got it absolutely right.
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"What a wonderful, wonderful finish from a good young player.
"That was what the game needed, it was almost looking like a stalemate."
Palmer is playing in his first game since winning the Under-21 European Championship with England, in which he played a major part in the winner.
It was his free-kick that deflected off Curtis Jones to hand England the victory and he has continued his fine form at Wembley.
On the grandest stage of all, Palmer stepped up with a beautiful goal that left Rasmdale with no chance.
It came about after another City academy prospect in Phil Foden left Thomas Partey for dead as he spun the Ghanaian and ran toward goal from midfield.
Kevin de Bruyne popped the ball off to Palmer, whose curling strike gave City the lead.
But Leandro Trossard's last-gasp strike, which took a heavy deflection, meant the match went to penalties.