Charlie Austin's 75th-minute penalty secured the points for Harry Redknapp's side, the striker sending Wayne Hennessey the wrong way to maintain the only unbeaten run in the division.
Their advantage at the summit is now three points after Blackpool and Leicester City fought out a 2-2 draw at Bloomfield Road.
Tom Ince's last-gasp spot-kick goal ensured that the hosts earned a share of the spoils after Paul Konchesky's penalty and Andy King's strike had eclipsed Chris Basham's opener.
Scott Arfield and Sam Vokes put the visitors into a two-goal lead, and although Matt Smith pulled one back for Brian McDermott's side, Burnley held on to record their first win at Elland Road since 2004.
Individual performance of the day went to Birmingham City debutant Jesse Lingard after the Manchester United loanee scored all four goals in the 4-1 victory over struggling Sheffield Wednesday.
His afternoon was soured by his late substitution due to injury, but his team-mates were able to secure their second win of the season despite Kamil Zayatte's consolation goal for the visitors.
After starting the season strongly, Nottingham Forest were held to their second draw in a week against Doncaster Rovers.
Federico Macheda, on loan from Manchester United, scored twice for Paul Dickov's side, but Chris Cohen and Andy Reid earned Forest a 2-2 draw.
For the hosts their poor start to the season continues, and Jason Scotland's early equaliser was the only positive for David Flitcroft's side.
Reading's 3-1 win over Derby County at Pride Park was sealed by Nick Blackman's stoppage-time goal, while Brighton came from a goal down to beat Bolton at the Amex Stadium.
Millwall won the early kick-off on Saturday thanks to a Scott McDonald strike against Charlton Athletic, while Blackburn Rovers and Huddersfield Town battled out a goalless draw.
