Following defeats for top-four rivals Tottenham Hotspur, Arsenal and Manchester United on Saturday and Sunday, the Blues could have moved up to third in the table with victory on home soil.
All four goals on the night were scored in the opening 24 minutes, as N'Golo Kante and Gonzalo Higuain struck shortly after Jeff Hendrick had given the visitors the lead, only for Ashley Barnes to level up and earn a share of the spoils for his side.
A point takes Chelsea up to fourth place, level on points with Tottenham and one ahead of Arsenal, who both have a game in hand to play.
Maurizio Sarri made four changes to his starting lineup on the back of the victory over Slavia Prague here four days ago, a result that keeps Chelsea's campaign alive on two fronts, but it was one of those who retained his place in the side that had the first shot on goal.
Eden Hazard's corner was cleared back into his path and his stinging shot from a tight angle was too hot for Tom Heaton to handle, though his defenders were able to help him out.
The Belgian forward then turned provider by lifting the ball over the Clarets' defence into the path of Higuain, who in turn lifted it over Heaton, only for Ben Mee to hook the ball off the line.
Having been penned in for the opening eight minutes, Burnley scored from their first attacking move as Hendrick volleyed a half-cleared corner into the bottom corner from 20 yards, picking his spot through a sea of bodies with a brilliant technique.
Chelsea needed just six minutes and 14 seconds to turn a one-goal deficit into a one-goal lead, though, thanks to Kante's well-taken finish from 10 yards following some fine wing play from Hazard - turning Matt Lowton inside out - before Higuain thumped the ball past Heaton via the underside of the crossbar after exchanging passes with Cesar Azpilicueta for the Blues' second.
Just when Chelsea were closing in on a third goal, Burnley once again struck against the run of play through Ashley Barnes, who was left unmarked in front of goal to convert a free kick after both Mee and Chris Wood had helped it into his path.
The sides shared a chance apiece in the opening 15 minutes of the second period - Higuain guiding a first-time shot on target for Heaton to make a routine save and Wood failing to get a shot away when played through on target at an angle.
The cutting edge shown by the hosts in the opening quarter of the match deserted them, however, with Azpilicueta's 15-yard drive, forcing Heaton into another straightforward save, the best they could muster in the remainder of the match.
There was some late drama as Sarri was ordered away from the dugout after reacting angrily to a decision that went against his side, sparking a coming together between both benches when the full-time whistle sounded.
CHELSEA (4-3-3): Kepa; Azpilicueta, Luiz, Christensen, Emerson; Kante (Kovacic 46'), Jorginho, Loftus-Cheek; Hudson-Odoi (Pedro 41'), Higuain (Giroud 77'), Hazard
BURNLEY (4-4-2): Heaton; Lowton, Tarkowski, Mee, Taylor; Hendrick, Westwood, Cork, McNeil; Barnes, Wood