Kai Havertz had put the hosts in front with a well dispatched header but, despite creating a lot of openings, Chelsea couldn't manage to kill the game off completely.
Both teams stay in 1st and 18th-place respectively, but it is a result that could prove to be vital at the end of the season. One or two points could be the difference between a Premier League title for one team and top-flight survival for the other.
Underneath are four talking points from the clash.
Thomas Tuchel was very frustrated in the aftermath of that game and it's easy to see why.
The Blues dominated the match from the first minute until the last, strangling the life out of Burnley. They switched off for a couple of seconds and Burnley remarkably made them pay.
Chelsea fans were downbeat and journalists questioned what went wrong but, in fairness, it was a near-perfect illustration of how to totally dominate and control a match. 99 times out of 100, Chelsea win that match. They had plenty of chances to kill the game but nobody was able to score that vital second.
It's worth remembering that Tuchel will have a refreshed Romelu Lukaku to call upon after the Belgian returns from injury. Games like this were tailor-made for the £97-million man to make the difference.
Pope clutch performance
The 29-year-old made a string of fine saves to keep out a rampant Chelsea team and those stops helped his team to secure a vital point at one of the toughest stadiums around.
In truth, Pope hasn't quite been at his brilliant best since recovering from his shoulder injury at the start of last season. While he hasn't been at fault for too many goals conceded, he hadn't pulled off enough clutch performances when his team have needed him to.
Reece James has had an up-and-down season but his performances of-late have been sensational; there was no let-up on Saturday afternoon.
Chelsea's wing-backs are always given license to roam but James was consistently one of the furthest players up the pitch. N'Golo Kante and Andreas Christensen could cover the space in-behind, allowing the academy star to basically play in a free-role in the attacking third.
Charlie Taylor actually put in a valiant effort down his side but it was an incredibly difficult afternoon. He had to contend with the tricky Callum Hudson-Odoi and this opened up more spaces for James to slip into and cause damage as the spare-man.
When Cesar Azpilicueta is preferred in that position, we see Ben Chilwell almost operating as a second-striker. Tuchel's system seems to grant one wing-back complete freedom while the other needs to be a little more disciplined. It's tough to argue giving James that freedom when he's in the form that he is.
Sean Dyche couldn't have written a much better script for himself after going a goal down to the Champions of Europe. The home fans were frothing at the mouth as they anticipated an onslaught and the Burnley travellers were probably fearing the worst.
However, the Burnley manager's two substitutes, Jay Rodriguez and Vydra, combined wonderfully to punish a profligate Chelsea attack in the latter stages. If Carlsberg did managerial team-talks!
In fairness, it was much about the collective resolve and mental strength as it was about the contributions of the substitutes. Vydra's goal would have meant a lot less if it wasn't for the strong jaw that was demonstrated in the 78 minutes prior to that action.
Burnley had been notorious for their defensive resilience and discipline but that hasn't always been so easy to see this year. Dyche will hope that the heroic efforts at Stamford Bridge will provide the catalyst for some much needed clean sheets later down the line.