Sarri's side fell behind to Jeff Hendrick's fine finish before N'Golo Kante equalised moments later.
In a frantic opening 24 minutes at Stamford Bridge, Gonzalo Higuain put Chelsea ahead, only for Burnley's Ashley Barnes to punish more dismal defending from the Blues.
Without a win in their last two league games, Chelsea will take only slender comfort from climbing one point above fifth placed Arsenal into fourth place, level on points with third placed Tottenham.
Chelsea would have gone third with a victory, but instead they have now played a game more than top four rivals Tottenham, Arsenal and Manchester United, who are three points behind in sixth.
It was a missed opportunity after defeats for Tottenham, Arsenal and United at the weekend and Chelsea were booed off at full-time.
In a fitting coda to a damaging evening, Sarri was sent to the stands for protesting in the final seconds before several members of the Blues bench were involved in a tetchy touchline clash with their Burnley counterparts.
Sarri's assistant Gianfranco Zola said his boss was so "emotional" after the game that he couldn't fulfil his media duties.
"He was very upset because he has heard not good words towards him from the Burnley bench. The only reason he walked over there was to get our players to go into position, he was trying to help the ref," Zola said.
- 'Too much time wasting' -
Zola took a swipe at Burnley's perceived gamesmanship, which included Clarets keeper Tom Heaton being booked in the first half for time-wasting.
"We are unhappy. There was too much time wasting. Five minutes injury time wasn't enough to compensate. We couldn't build important rhythm," Zola moaned.
"We expected it to be a tough game. We didn't expect so much time wasting. We expected (referee) Kevin Friend to give more extra time. That's why we're very unhappy."
Chelsea defender David Luiz went even further in his criticism of Burnley.
"It's anti-football. Wasting time all the time, especially when you have the ball. Their players went to the floor and stopped the game," he said.
Now Arsenal can replace Chelsea in fourth if they avoid defeat at Wolves on Wednesday.
"We need to improve but we're in a good way. We care, as we showed today," Zola said.
Adding to Sarri's problems, he was hit by first half injuries to France midfielder Kante and England winger Callum Hudson-Odoi.
"It doesn't look very good unfortunately," Zola said of Hudson-Odoi. "It's a problem with his Achilles tendon."
Brushing off the bench-clearing clash, Burnley boss Sean Dyche said: "By the time I turned round there was something going off in the tunnel.
"I don't know what you can call it now. Handbags? Manbags? Bumbags?"
Chelsea were stunned in the eighth minute when Hendrick's rocket put Burnley ahead.
It was a poor goal for Chelsea to concede as Cesar Azpilicueta could only head a Burnley corner to the edge of the area, where Hendrick was left unmarked to volley a superb dipping strike past Kepa Arrizabalaga.
- Slow to react -
Eden Hazard's majestic assist helped Chelsea draw level four minutes later, underlining why the club are so keen to keep their talisman out of Real Madrid's clutches.
Cutting in from the left, Hazard twisted and turned so rapidly that Burnley defender Matt Lowton fell flat on his backside as the Belgian picked out Kante for a powerful shot that flashed past Heaton from 10 yards out.
Pushing forward with pace and precision, Chelsea went in front two minutes later when Higuain turned smartly to pick out Azpilicueta.
The Chelsea defender executed a deft back-heel to Higuain, who continued his run and crashed a brilliant finish into the roof of the net from an acute angle.
But Burnley punished more poor defending to equalise in the 24th minute.
With Chelsea slow to react to a Burnley free-kick, Ben Mee's header was glanced on by Chris Wood and the unmarked Barnes apply the finishing touch with a close-range volley.