George Boyd gave the Clarets the lead with a scintillating strike from 22 yards in the 34th minute, but the Magpies levelled three minutes after the break when Papiss Cisse kneed home the equaliser for his fifth goal of the season.
Newcastle drop down to tenth.
Manchester United's recent revival stretched to a fourth successive victory to hand former old boy and Stoke boss Mark Hughes a third straight defeat.
It was two players with a combined worth of £64.6million that lifted boss Louis van Gaal's side to within eight points of Barclays Premier League leaders Chelsea with goals in a 2-1 win over the Potters.
Marouane Fellaini, who cost United £27.5million upon his arrival from Everton in September last year, headed the Reds into a 21st-minute lead in meeting a cross from Ander Herrera.
City, who have now lost on their last 12 visits to Old Trafford, were back on level terms six minutes before the break when Steven N'Zonzi rifled home a fizzing shot from just outside the area.
But it was Juan Mata, a £36.1million capture from Chelsea in January, who struck the decisive blow with a free-kick from wide on the right that should have been cleared, only to drift into the net.
Defender Marcos Rojo appeared to indicate he had got a touch to the ball on its way in, but Mata claimed the winning goal.
Liverpool further eased the pressure on manager Brendan Rodgers with a 3-1 victory at rock-bottom Leicester, following on from their own win over Stoke on Saturday.
Goalkeeper Simon Mignolet has come in for considerable criticism of late, and it was the Belgian who inadvertently gave City the lead in the 22nd minute when a Leonardo Ulloa struck a post, rebounded off his back and over the line.
But Liverpool fought back with goals from Adam Lallana and Steven Gerrard, the latter restored to the side, before Wes Morgan was sent off for a foul on Rickie Lambert in the 63rd minute.
Jordan Henderson then sealed the points in the 83rd minute to hand Leicester their seventh defeat from their last nine matches.
Swansea are up to sixth following a 2-0 win at the Liberty Stadium over relegation-threatened QPR.
The Swans left it late, however, as Ki Sung-yueng netted his first goal since the opening day of the season in the 78th minute.
That was followed five minutes later by former Rangers midfielder Wayne Routledge clinching the victory with a strike from outside the box.
Barclays Prem
Leicester 1 - 3 Liverpool
Burnley 1 - 1 Newcastle
Man Utd 2 - 1 Stoke
Swansea 2 - 0 QPR
W Brom 1 - 2 West Ham
C Palace 0 - 1 A Villa
West Ham continue to ride high in fifth after they came from behind to beat West Brom 2-1 at The Hawthorns.
The Baggies had taken a 10th-minute lead when Craig Dawson headed in a free kick from Graeme Dorrans, ending the home side's goal drought at 323 minutes.
But first Kevin Nolan pounced after goalkeeper Ben Foster had pushed away an overhead kick from Andy Carroll in the 35th minute prior to centre-back James Tomkins scoring his first goal for two years on the stroke of half time.
Aston Villa are up to 12th after Christian Benteke scored the only goal of the game in a vital win for Paul Lambert's side at Crystal Palace.
Making only his fifth appearance of the season due to injuries, Benteke capitalised on a mistake from Scott Dann inside his own half before striding forward and confidently beating Julian Speroni for his first goal since March.