Cole Palmer scored twice for the Blues, but his first effort was cancelled out by Josh Cullen, before Dara O'Shea came up with a late leveller for the visitors, who were reduced to 10 men in the latter stages of the first period and also had manager Vincent Kompany sent off.
Burnley have moved onto 18 points but still sit 19th in the table, four points from the safety of 17th position, while Chelsea remain 11th, boasting 40 points from their 28 matches this term.
Chelsea were on the front foot in the early stages of the contest, with Palmer striking one into the side-netting after a bright move from the home side.
Burnley responded well in the period that followed, but they were almost caught out by a deflected strike from Enzo Fernandez in the eighth minute, which Arijanet Muric needed to tip over the crossbar.
Palmer then fired wide of the target twice in quick succession, before Burnley had an excellent opportunity of their own, with Wilson Odobert seeing his shot pushed behind by Chelsea goalkeeper Djordje Petrovic.
Nicolas Jackson was next to try his luck for Chelsea, with his attempt being kept out by Muric, but the Blues did have the ball in the back of the Burnley net in the 20th minute.
Axel Disasi appeared to head a Mykhaylo Mudryk cross into the back of the net, but a resulting VAR check demonstrated that the defender had used his arm, so it was disallowed for handball.
Burnley were living dangerously at the back, but they were carrying a threat down the other end, and Petrovic had to be alert to keep out a strike from Cullen in the 26th minute.
Mudryk then tried his luck for Chelsea, with the Ukraine international drawing a smart save from Muric after smart work from Jackson in the final third of the field.
Soon after, Chelsea were awarded a penalty, with Mudryk pushed to the ground by Lorenz Assignon, who was then shown a second yellow card, reducing the visitors to 10 men in the process.
Burnley boss Kompany was sent off for his protest against the decision, but VAR agreed with the on-field call, and Palmer stepped up to send Chelsea ahead in the 44th minute.
Chelsea came close to doubling their lead in the final stages of the first period through Malo Gusto, but Burnley goalkeeper Muric was on hand to make a smart save.
Burnley incredibly managed to level the scores in the 47th minute through Cullen, who expertly picked out the corner of the net with an excellent half-volley from the edge of the Chelsea penalty box.
Petrovic made a smart save to keep out a header from Lyle Foster in the period that followed, before Muric kept out an effort from Jackson down the other end, with both teams pushing for a second at Stamford Bridge.
Muric was again called into action to deny Noni Madueke, but Chelsea did manage to regain the lead in the 78th minute through Palmer, with the Englishman netting his second of the match, picking out the bottom corner from inside the box after being found by Raheem Sterling.
Burnley were not finished, though, and levelled the scores three minutes later, with O'Shea heading into the back of the net from a corner to spark wild celebrations in the away end.
Both teams had chances to win it in the latter stages of normal time, but Sterling headed wide of the target, before Jay Rodriguez thumped a header against the crossbar down the other end of the field, with the points ultimately being shared at Stamford Bridge.