The Blues came into the game boosted by their top-four rivals Arsenal, Manchester United and Tottenham all losing over the weekend, although Burnley arrived on their best run of the season having defeated struggling Cardiff last weekend.
The Clarets won the corresponding fixture on the opening day of last season and they went ahead inside the opening 10 minutes through Jeff Hendrick.
Chelsea responded imminently with two goals in as many minutes from N’Golo Kante and Gonzalo Higuian putting the hosts ahead before the 15-minute mark.
However, the visitors hit back to level midway through the half with Ashley Barnes on target.
Chelsea dominated the rest of the encounter but could not find an elusive winning goal and have to settle for a point which takes them above Arsenal and level on 67 points with third-placed Tottenham.
The draw moves Burnley on to the magic 40-point mark and virtually guarantees them another season of Premier League football.
Chelsea made four changes from the XI which beat Slavia Prague 4-3 at Stamford Bridge in the Europa League on Thursday night as Jorginho, Ruben Loftus-Cheek, Callum Hudson-Odoi and Higuain replaced Ross Barkley, Mateo Kovacic, Pedro and Olivier Giroud.
Burnley were unchanged for the third consecutive game as Sean Dyche's side were targeting a fourth successive Premier League win.
Chelsea started strongly and almost went in front in the sixth minute when Eden Hazard played in Higuain who lifted the ball over Tom Heaton from an acute angle only to see Ben Mee acrobatically hook it clear off his own line.
Burnley then took the lead with their first shot of the game on eight minutes when a corner drifted out to Hendrick on the edge of the box and he hit a sweet first-time volley into the top corner past Kepa Arrizabalaga.
Chelsea responded positively and were level within four minutes when Hazard turned Matt Lowton inside out down the left before putting the ball on a plate for Kante to rifle in a fierce effort past Heaton.
Two minutes later the comeback was complete when Higuain exchanged passes with Cesar Azpilicueta down the right of the Clarets area before hitting an unstoppable effort from the angle which flew into the net off the underside of the crossbar.
Chelsea continued to dominate, but they were pegged back on 24 minutes when a free-kick into the area was headed back into the six-yard box by Mee at the back post towards Chris Wood and he then teed up Barnes for a simple tap-in.
The goalmouth action seemed to dry up after the equaliser and Chelsea were hit by the loss of Hudson-Odoi who had to be helped off on 40 minutes with what looked like a muscle injury.
There was a slower pace to the start of the second half with Higuain having the first attempt of note on 52 minute swhen he got on the end of a Pedro pass in the middle of the area but saw his low strike go straight at Heaton.
Heaton was relieved on 58 minutes when Emerson picked up possession down the left channel and tried his luck from 25 yards but the ball curled just wide of the far post.
Chelsea continued to press Burnley back and they almost caught the visitors out in the 67th minute when Emerson’s cross dropped to Azpilicueta at the back post and his effort was beaten away by Heaton.
Despite enjoying the vast majority of possession, the hosts struggled to find a way past the resolute Clarets’ rearguard.
Kovacic did managed to put in a dangerous cross towards the back post which Loftus-Cheek poked back into the middle but Olivier Giroud was unable to get a decisive touch.
The frustration got the better of Sarri in stoppage time and he was sent to the stands, before the injured Antonio Rudiger got involved in a heated exchange with the Burnley coaching staff after the final whistle.