Having gone close through Enzo Fernandez and Nicolas Jackson – as well as seeing Axel Disasi denied by VAR – Chelsea broke through just before half-time.
Assignon was dismissed for a second yellow card after bundling over Mykhailo Mudryk in the box, before Palmer’s delightful Panenka from the penalty spot put the hosts in front.
Burnley responded within two minutes of the restart. Cullen played a neat one-two with substitute Josh Brownhill, and drilled a tremendous 20-yard volley past Djordje Petrovic.
Palmer restored Chelsea’s lead in the 78th minute, latching onto Raheem Sterling’s neat flick before slotting past Arijanet Muric.
However, the visitors hit back four minutes later as O’Shea headed in another equaliser.
Both sides went close to snatching all three points late on. Sterling nodded wide from close range, while Jay Rodriguez headed against the crossbar as the points were shared at the Bridge.
WHAT DOES IT MEAN? WASTEFUL CHELSEA FEEL THE BLUES
Following West Ham’s collapse at Newcastle earlier on, Chelsea had the chance to close the gap on the seventh-place Hammers to just two points – and still with two games in hand.
The Blues controlled large periods of the contest with 69% of the possession, while they twice led and also had 33 shots on goal, but to no avail.
Burnley also registered 18 attempts with the combined total of 51 the second-highest in a single Premier League game on record (2003-04), after Queens Park Rangers’ showdown with Leicester City in November 2014 (52).