The Blues were on the verge of being made to pay for profligacy when Michael Olise cancelled out Mykhaylo Mudryk's opener, but with one minute of normal time remaining, Noni Madueke won and converted a penalty to help the Blues on their way to a fourth successive home victory in all tournaments.
History had already been written for Chelsea before kickoff, as a fledgling XI - comprising the promoted Christopher Nkunku and Ian Maatsen - had an average age of just 23 years and 21 days, their youngest ever starting lineup in the Premier League.
While Palace made the brighter start, Chelsea soon began to force the issue and came within inches of taking the lead in the ninth minute through Maatsen, whose shot squirmed under Dean Henderson before Tyrick Mitchell produced an astounding burst to clear the ball off the line.
However, Mauricio Pochettino's side - and Mudryk - were not to be denied their rightful opener in the 13th minute, as Nkunku failed to slip the Ukrainian through but picked out Malo Gusto on the left, and the full-back cut back for Mudryk to tap in from close range.
A slick turn from Gusto earlier in the move and a Nathaniel Clyne slip were both pivotal in Chelsea's opener, and the effervescent Mudryk was presented with a golden chance to double his tally in the 22nd minute, only for Henderson to close down the angle after a neat Nicolas Jackson flick set him away.
Pochettino's side remained in the ascendancy as the half drew to a close, albeit without busting a gut to try to add to their advantage - much to the Argentine's fury - and they were hit by an Eagles sucker punch in the first minute of injury time.
Jordan Ayew aimed to pick out Jean-Philippe Mateta with an inswinging cross from the left, which sailed over the head of the Frenchman and Levi Colwill - who unsuccessfully tried to clear the danger - and an unmarked Olise produced a sublime piece of chest control before volleying in at Dorde Petrovic's near post.
Rocked by Olise's unscripted response, Chelsea also found themselves on the back foot in the early stages of the second half and were powerless to prevent Mateta from surging through and unleashing a strike in the 55th minute, but Petrovic covered his near post well.
Pochettino's next plan of action was to bring on Romeo Lavia for his belated Blues debut, while Madueke was also introduced and produced a terrific burst of pace to get away from Chris Richards before being pulled back by the Palace man; referee Michael Salisbury erroneously brandished the red cad before quickly realising his mistake.
From the resulting free kick - which was taken quickly - a lovely passing sequence ended with Gallagher slipping Jackson in, but the Senegalese added another horror miss to his collection, curling wide while one-on-one with Henderson in the 74th minute.
Jackson had seemingly made amends just two minutes later, meeting Thiago Silva's deflected cross with an instinctive first-time toe poke to prod Chelsea ahead, but he had failed to beat the offside trap and the score remained 1-1.
After setting up a disallowed goal at the other end of the pitch, a slip from Silva almost saw him inadvertently turn provider for Palace in the 81st minute, but Olise was denied by Petrovic from a tight angle and the resulting set piece came to nothing.
The Eagles then failed to make the most of a four-on-two counter-attack after Salisbury waved away Chelsea appeals for a penalty, but upon review, Eberechi Eze had clearly tripped Madueke just inside the area, and the Blues were awarded their rightful spot kick.
Madueke dusted himself down to step up to the 12-yard mark, and after a stuttering run-up, the 21-year-old sent Henderson the wrong way to restore Chelsea's slender advantage in the 89th minute.
Roy Hodgson's side were granted eight minutes of additional time to try to respond again, but after one final saved Olise effort and a pair of bookings for Pochettino and skipper Conor Gallagher, the hosts' three points were safe.
By virtue of a 12th successive Premier League win over Crystal Palace, Pochettino's side have risen above Bournemouth into 10th place in the table, while Palace will stay 15th unless Everton can claim a shock win over Manchester City in the late kickoff.
Chelsea and Palace both have one final engagement for 2023, as the Blues head to Luton Town for Saturday's early kickoff, a couple of hours before the Seagulls host Brentford in another capital clash.