Burnely were first to find the back of the net through Jeff Hendrick but the Blues responded with two quick goals from N’Golo Kante and Gonzalo Higuain.
However, the visitors were back on level minutes later through Ashley Barnes. The Blues pushed for a winning goal but the Clarets held to secure one point.
The draw moves Chelsea to fourth place in the league table.
Amid the chaos, Burnley celebrated, having dug in and fought hard for the draw which took them to 40 points and all but ensures they will playing Premier League football for a third successive season.
Chelsea grumbled about 'anti-football' as they squandered the opportunity to put pressure on their top-four rivals after a weekend when Tottenham, Arsenal and Manchester United had all lost.
A point is added to the total and they remain in the hunt for the Champions League but this setback was deepened by concerns over an injury to Callum Hudson-Odoi, who was forced off in the first half with an achilles injury.
-Dailymail