Clarets striker Danny Ings scored his 11th goal of the season in all competitions in the 21st minute, firing in from Kieran Trippier's pass.
The hosts extended their advantage through Sam Vokes' eighth goal of the campaign in the 80th minute and while Jason Shackell's own-goal two minutes later made for a nervy finale, the home side managed to hold on.
Second-placed QPR remain level on points with the leaders with a game in hand following their 2-0 win over rock-bottom Barnsley at Loftus Road.
Former Burnley man Charlie Austin netted both goals, his first strike coming in the 66th minute and his second, a penalty, in the 87th.
Third-placed Leicester lost ground on the top two, having been beaten 1-0 at Doncaster.
An own-goal from Foxes goalkeeper Kasper Schmeichel in the 17th minute proved decisive, leaving them three points off the top two having played the same amount of games as Burnley.
Watford surged into the play-off places after winning 2-1 at Huddersfield, the away side coming from behind with goals from Fernando Forestieri and Daniel Pudil following Daniel Ward's opener.
Bolton moved out of the relegation zone after securing a 2-1 victory at Birmingham in the day's early kick-off - their first league win of the season.
Jermaine Beckford and Neil Danns scored for the visitors before Nikola Zigic's reply.
That sent Sheffield Wednesday into the bottom three and the Owls, still looking for their maiden victory of the campaign, stayed there after drawing 1-1 at home with Ipswich, falling behind early on to a Paul Anderson goal that was swiftly cancelled out by Michail Antonio's leveller.
The other side in the relegation zone are Yeovil, who were thumped 4-1 at Middlesbrough.
The visitors led through an early Liam Davis goal, but Grant Leadbitter soon equalised and when his penalty was saved, Albert Adomah stabbed home. Jacob Butterfield and Kei Kamara added goals after the break.
Elsewhere, Blackpool drew 0-0 at Charlton, while Derby, with Steve McClaren in charge for the first time, beat Leeds 3-1 at home.
The Rams went 2-0 up through Chris Martin and Johnny Russell and after Jason Pearce pulled a goal back, Will Hughes wrapped up the victory.
The other 3pm kick-off saw Bournemouth stage an impressive comeback at home against Millwall, running out 5-2 winners.
The Lions had a two-goal advantage inside the first 10 minutes through Martyn Waghorn and Liam Trotter, but Ryan Fraser made it 2-1 just before half-time and five minutes after the break, Steve Cook drew things level with a header.
That was the first of three goals in nine minutes as Harry Arter put Bournemouth in front and Lewis Grabban then converted a penalty - awarded for a handball by Alan Dunne, who was sent off - before Brett Pitman tucked away another spot-kick in stoppage time.