Kai Havertz's first-half opener was canceled out by Matej Vydra in the 79th minute as the relegation-battling Clarets earned an unlikely point.
Still without a natural striker, the Blues were dominant in the match and had the visitors on the back-foot for most of the game.
But a lack of cutting edge kept them from extending their lead, and their wastefulness eventually came back to haunt them late on.
Manchester City won against Manchester United earlier in the day. Chelsea's lead at the top is now three points but Liverpool can trim it down to just one with a victory at West Ham on Sunday.
Here are the five talking points from the match:
#1 Chelsea lacked killer instinct despite attacking dominance
With 25 shots in the match and enjoying 70% of the possession, Chelsea created enough chances to easily win the contest by three or four goals. But without Romelu Lukaku and Timo Werner, they lacked that killer instinct to put the match to bed.
Also, Mason Mount was only fit enough to start on the bench.
Credit also goes to Burnley keeper Nick Pope, who made three fine saves to frustrate the Blues and keep his side in the game. It obviously paid dividends as they ultimately walked away with a point.
#2 Burnley end Chelsea's open-play hoodoo
So far, Chelsea had conceded only thrice in the league - two penalties and one goal from a corner. No side had managed to score from open play, despite almost a third of the season in.
Burnley, mired in a relegation scramble and scoring only 10 times entering the game, were the least likely to score. But they tore the script apart by netting one from open play late on.
Matthew Lowton hooked a cross for Jay Rodriguez, who unselfishly headed it into the path of Vydra. The Czech Republic forward calmly slotted past Mendy, whose run of four consecutive league clean sheets came to an end.
#3 Reece James delivers again
Reece James is having quite the season for Chelsea. With four goals and three assists in the league, he's posting numbers that a forward would be proud of, let alone a full-back. He's currently Chelsea's leading goalscorer in the Premier League this season with four goals.
His assist for Havertz today was a class act, sending a looping ball for the German to nod home from close range. James wasted no time in proving his worth to the side on his first-team return.
#4 Havertz, a possible No.9?
In the last four games that he's started as their striker, Havertz has scored twice and made one assist. This includes tonight's goal, demonstrating his No.9 credentials.
He started as their makeshift centre-forward and once again repaid Tuchel's faith with an important goal. It was a fine header between two of Burnley's centre-backs.
With the international break coming up, Werner and Lukaku will have ample more time to recover. They might as well return for the away clash with Leicester City.
However, the German's good showings in the striker's role have given Tuchel plenty of food for thought.
#5 Chelsea's title aspirations face defining weeks ahead
Chelsea have been on a good run of form this season, with many even touting the reigning European champions to win their first league title since 2017.
However, following today's draw, they're only three points clear of second-placed City. Liverpool could close the gap to just one if they beat West Ham on Sunday.
So there's still no clear winner in the title race just yet. The Blues, especially, have a tough run after the break, with games against Leicester, Watford, Manchester United and West Ham coming in quick succession.
Defining period ahead for Tuchel and co.