Jose Mourinho's leaders were held 1-1 at Southampton earlier in the day but City failed to take advantage of that result after failing to keep a two-goal lead against the Clarets.
Chelsea had won five successive games in all competitions but they were behind at St Mary's after 17 minutes when Sadio Mane broke on to Dusan Tadic's pass and away from John Terry to deftly chip over Thibaut Courtois.
Eden Hazard, excellent throughout, produced a brilliant equaliser on the stroke of half-time when he slotted powerfully into the far corner.
Chelsea should have had a penalty when Cesc Fabregas tumbled under the challenge of young Saints defender Matt Targett - making his first Premier League start - but referee Anthony Taylor added insult to injury when he booked the Spaniard for simulation.
The Blues could not find a winner from late pressure even when Saints midfielder Morgan Schneiderlin was sent off two minutes from time for a second bookable offence.
Manuel Pellegrini's City seemed on the way to a club record 10th successive win when David Silva and Fernandinho struck before half-time.
But George Boyd pulled one back for the Clarets when he turned home Danny Ings' shot despite looking suspiciously offside.
Ashley Barnes stunned the Etihad Stadium by reacting quickest in a crowded penalty area to hand Burnley a precious point in their battle against relegation.
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Tottenham and Manchester United produced an entertaining goalless draw at White Hart Lane after Louis van Gaal's side had dominated the first half.
Hugo Lloris made stunning saves from United trio Radamel Falcao, Robin van Persie and Ashley Young while Juan Mata's deflected free-kick struck a post and Phil Jones had an effort disallowed for offside.
Spurs, who had not beaten United at home since 2001, improved after the break and David de Gea was forced into a couple of saves before the game petered out tamely.
Arsenal were unbeaten in their last 12 league games against West Ham and they kept that fine run going with a 2-1 win to leapfrog the Hammers and move into fifth spot.
Santi Cazorla put Arsenal ahead from the penalty spot four minutes before the break after he had been fouled by Winston Reid and Danny Welbeck added a second moments before the interval.
A Villa 0 - 0 S'land
So'ton 1 - 1 Chelsea
Tottenham 0 - 0 Man Utd
Hull City 0 - 1 Leicester
Man City 2 - 2 Burnley
Newcastle 3 - 2 Everton
West Ham 1 - 2 Arsenal
Stoke 2 - 0 W Brom
QPR 0 - 0 C Palace
Cheikhou Kouyate gave West Ham hope with a 54th-minute header but Arsenal held on for a crucial three points at Upton Park.
Papiss Cisse's fifth goal in six games launched a Newcastle fightback as they ended a four-match losing streak with a 3-2 victory over Everton.
The Senegal international cancelled out Arouna Kone's fifth-minute opener - his first for Everton on his opening Barclays Premier League start for the club he joined during the summer of 2013 - with a close-range finish 11 minutes before the break.
However, Toffees boss Roberto Martinez may feel the forward was lucky to still be on the pitch after appearing to catch full-back Seamus Coleman with a flailing arm four minutes earlier, and there may yet be repercussions with referee Craig Pawson having taken no action at the time.
But if it was Cisse who started the ball rolling, it was Ayoze Perez and Jack Colback who completed the job with the Spaniard's fifth goal for the club giving them a 51st-minute lead which was cemented by Colback's first in a Newcastle shirt 16 minutes later, although substitute Kevin Mirallas' late strike made for a tense conclusion.
Alan Irvine's tenure as West Brom manager is looking precariously thin after the Baggies slid to a 2-0 defeat at Stoke.
Albion's seventh defeat in nine games was confirmed after the break as Mame Biram Diouf struck twice for his fifth and sixth goals of the season but his first success in front of goal since the start of November.
Goal-shy Aston Villa drew another blank in the 0-0 draw at home to Sunderland having had midfielder Fabian Delph sent off after 48 minutes.
Delph saw red for a tackle on Sunderland's Jordi Gomez but Villa, the lowest scorers in the division with 11 goals from 19 games, held on for a point.
Managerless Crystal Palace managed a goalless draw at QPR the day after Neil Warnock's sacking at Selhurst Park.
The draw does little for either team, with QPR in 15th place and Palace in 18th and remaining in the relegation zone on goal difference.
Bottom-of-the-table Leicester managed their first win in 14 attempts since beating Manchester United 5-3 at the end of September.
Riyad Mahrez scored the all-important 32nd-minute goal in the 1-0 victory at Hull and Leicester had to play the final three minutes with 10 men after Paul Konchesky was sent off for two bookable offences.
Hull's Stephen Quinn evened the numbers up in injury time with his red card as Leicester moved within three points of safety.