Maurizio Sarri's men were hoping to bounce back from a 2-0 defeat to Liverpool in their previous top-flight outing but they dropped points once again and could be facing key games against Manchester United and Eintracht Frankfurt without Kante.
Burnley – still yet to confirm their survival – went ahead through Jeff Hendrick early on, and although Kante and Gonzalo Higuain saw Chelsea turn things around in the space of two minutes, Ashley Barnes pegged them back in an enthralling first half.
Chelsea did much of the attacking after the interval but Sarri – who was removed from the sidelines in stoppage time – saw his side fail to make the most of top-four rivals Tottenham, Arsenal and United losing at the weekend.
Burnley edged ahead just eight minutes into a thrilling first half when Hendrick sent a fine volley into the bottom-right corner, but the Blues swiftly equalised thanks to Kante's left-footed effort into the roof of the net after fine work from Eden Hazard.
A brilliant move in the 14th minute saw Higuain score for the first time since March 3, rifling in off the crossbar from a tight angle following Cesar Azpilicueta's clever flick.
However, Burnley restored parity 10 minutes later when Barnes latched on to Chris Wood's glancing header.
PICS: Clarets at the double! pic.twitter.com/j6iL29YtQW— Burnley FC (@BurnleyOfficial) April 22, 2019
Having lost Hudson-Odoi late in the first half to an apparent shin problem, Kante was unable to come out for the second period due to a reported back injury.
Chelsea still looked cohesive going forward and nearly retook the lead in the 51st minute, but Tom Heaton thwarted Higuain – who reacted furiously when he was replaced by Olivier Giroud in the 77th minute.
The hosts piled the pressure on towards the end but failed to craft any further opportunities and Sarri's late ejection for dissent and a fracas after the final whistle – which was met with boos – summed up Chelsea's day.
What does it mean? Top-four race tighter than ever
A victory would have put Chelsea on 69 points, two clear of Tottenham, three ahead of Arsenal and five away from United.
With their rivals all in action on Tuesday or Wednesday, Chelsea could instead find themselves in an even tighter tussle if results do not go their way.
Although Hazard impressed once again for Chelsea, the standout player on the day was Burnley defender Mee. While he was turned a little too easily by Higuain for the Argentinian's goal, the centre-back recovered well and was excellent in the second half, making 10 clearances and three interceptions across the 90 minutes.
Emerson a disappointment on the left
While few players had obviously poor outings, Emerson Palmieri was underwhelming on Chelsea's left side. The hosts were on the front foot for most of the game, but the attack-minded full-back failed to offer much quality, with only one of his nine crosses finding a team-mate.
Key Opta Facts
4 - N'Golo Kante has scored four goals in 34 Premier League games this season, one more than he netted in his previous three campaigns in the competition combined (three goals in 106 apps). Advanced. #CHEBUR pic.twitter.com/cRP9a1H9Co— OptaJoe (@OptaJoe) April 22, 2019
- Chelsea conceded two goals from set-pieces (excluding penalties) in a single Premier League game at Stamford Bridge for the first time since May 2011 versus Newcastle United.
- Hazard has provided 13 assists in the Premier League in 2018-19, the most in the competition this season and one more than he registered in his previous three campaigns combined.
- Barnes has scored 11 Premier League goals this season, the joint-most by a Burnley player in a single campaign in the competition (also Danny Ings in 2014-15).
- Higuain has scored 10 or more top-flight goals in each of his last 11 seasons in Europe's top five leagues (four for Chelsea, six for Milan in 2018-19).
Chelsea travel to United for a potentially vital match in the top-four race on Sunday, while Burnley's quest to finally confirm their survival continues at home to pacesetters Manchester City.