The Chelsea forward broke one Premier League record and equalled another one as he equalised from 12 yards.
His strike also saw the Blues claw their way back to 2-2 after going 2-0 down.
Palmer levelled from the penalty spot after Yves Bissouma fouled Moises Caicedo.
The England international stepped up to send Fraser Forster the wrong way and extend his 100 per cent Premier League spot-kick record.
It extended his penalty record to 11 out of 11 and equal Yaya Toure for the joint-most penalties without missing in the division's history.
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Palmer then broke the record after Enzo Fernandez had put the visitors in front.
After being fouled by Pape Matar Sarr, he stepped up to chip the ball down the middle and move to 12 goals from 12 penalties in the Premier League.
The 22-year-old's first goal was also his 37th Premier League goal contribution of the year.
As a result, he has surpassed Jimmy Floyd Hasselbaink for the most by a Chelsea player in a single calendar year.
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The Dutchman held the record for 23 years, having set it back in 2001.
Across four years at Stamford Bridge, Hasselbaink scored 69 goals and provided 29 assists in just 136 appearances.
Meanwhile, Palmer's stunning record for the Blues now stands at 57 goal contributions in 62 matches across all competitions.
Enzo Maresca's side are also back into second place following their comeback win over Spurs.
Arsenal had earlier leapfrogged Chelsea by drawing at Fulham.
However, the Italian and his players closed the gap to leaders Liverpool to four points having played a game more.
The win also extends their unbeaten league run to six matches, dating back to a defeat at Anfield in October.