Palmer's 101st-minute strike was timed as the latest winning goal in Premier League history.
United struggled with the pace and dominance of Chelsea in the first half but staged a comeback with goals from Garnacho and Fernandes.
A breathless second half with goals from both sides set the stage for Palmer's late heroics.
Chelsea's swift attacking moves and clinical penalties put them in the lead, but United's fightback led to a stunning finale.
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