Chelsea, Aston Villa held in blow to Champions League hopes
The Blues climb into the Premier League's top four only on goal difference after they paid for failing to build on Joao Pedro's fourth-minute opener.
Chelsea were dominant until Wesley Fofana was sent off with 18 minutes remaining for a second yellow card.
Burnley took full advantage as Zian Flemming headed in from a corner in the 93rd minute to keep their slim hopes of survival alive.
The Clarets edge to within eight points of safety.
Villa defied the odds to keep pace with Arsenal and Manchester City in the title race until recent weeks, but a 1-1 draw continued their poor run at Villa Park.
Anton Stach's spectacular free-kick from fully 40 yards (37 metres) at an angle caught out Emi Martinez at his near post to give Leeds the advantage.
Daniel Farke's men have lost just twice in their last 16 games to relieve their fears of being dragged into a battle for survival.
But Tammy Abraham then snatched a point for Villa two minutes from time.
Five Premier League sides are set to qualify for next season's Champions League thanks to the strong performance of English sides in European competition this season.
But Liverpool and Manchester United have the chance to close in when they face Nottingham Forest and Everton in the coming days.
James Milner set a new record of 654 Premier League appearances in Brighton's 2-0 win at Brentford.
The 40-year-old started for the Seagulls as they secured a first win in seven games to ease the pressure on boss Fabian Hurzeler.
Diego Gomez drilled in the opening goal for the visitors after Ferdi Kadioglu's stunning strike came back off the bar.
Danny Welbeck then pounced on some slack Brentford defending to double Brighton's lead just before half-time.
Manchester City can close to within two points of leaders Arsenal when they host Newcastle in Saturday's 2000GMT kick-off.
The Gunners have breathed new life into City's title challenge with disappointing draws at Brentford and Wolves since Pep Guardiola's men last played in the Premier League.
Arsenal are in action on Sunday when they travel to local rivals Tottenham Hotspur in the north London derby.