The London derby at the Tottenham Hotspur Stadium started in the worst possible fashion for Enzo Maresca's men, who were two goals down inside the opening 11 minutes of Sunday's headline act.
However, after Spurs defender Cristian Romero was substituted with an injury, Chelsea quickly got themselves back into the tie through Jadon Sancho and threatened consistently before the half-time whistle.
Palmer and Sancho were often targeted by pieces of cardboard thrown from the crowd at corners, and Ange Postecoglou's men got to half time with their slender lead intact, only to implode in the second period.
A nonsensical challenge from Yves Bissouma on Moises Caicedo offered Palmer the chance to bring Chelsea level from the spot, which he did not pass up with a clinical strike into the bottom corner.
A sweetly-struck volley from Enzo Fernandez completed a captivating Chelsea comeback, before Bissouma's midfield partner Pape Sarr then copied his colleague with an inexcusable challenge from behind on Palmer.
Palmer 12 for 12 on Premier League penalties
The former Manchester City product stepped up again and pulled off an outrageous panenka penalty as Fraser Forster went to his right, before Son Heung-min's late strike gave Tottenham a flicker of hope.
However, Palmer's two penalties proved decisive, and the England international has now scored each of his first 12 spot kicks in the Premier League, breaking a competition record.
Yaya Toure held the previous high of scoring 11 straight Premier League penalties before fluffing his lines for the first time, but Palmer has now dethroned him as the penalty conversion king.
The 22-year-old's double has taken Chelsea to within four points of Premier League leaders Liverpool in second place, as Maresca's team capitalised on both Arsenal and Manchester City dropping points over the weekend.
The Blues now prepare to travel to Astana in the Conference League on Thursday night before hosting Brentford in a top-flight West London derby next Sunday.
Written by
Ben Knapton
Valencia 
Alaves 
PSV 
Feyenoord 
Bordeaux 
Rennes 
ST Etienne 
Marseille 
Monaco 
Everton 
Chelsea 
Leicester 
Wolverhampton 
Manchester Utd 
Bournemouth 
Real Madrid 
Sevilla 
Cremonese 
Sampdoria 
Tottenham 
Liverpool 
VFL Wolfsburg 
Borussia Dortmund 
Las Palmas 
Espanyol 
Monza 
Juventus 
Lens 
Paris Saint Germain 
Fiorentina 
Udinese 
Benfica 
Estoril 
Inter 
Como 
Manchester City 
Fulham 
Nottingham Forest 
Celtic 
Motherwell 
ST Mirren 
Rangers 
Carrarese 
Arsenal 
Ipswich 
Cagliari 
Lazio 
Atalanta 
FC Porto 
Boavista