Sam Vokes pounced late on for Sean Dyche's side to move them into sixth spot with 19 points -- the same as Chelsea, who host Manchester United on Sunday, and Arsenal.
Burnley have already won three league matches on the road this season compared to one in the whole of last season.
"I am delighted to be where we are but there are a lot of challenges coming our way and the group know we will have to work hard for every point we get," Dyche, who has been linked with the vacant Everton manager's job, told the BBC.
"We still have to fight for every chance we get but we are willing to do the graft."
Liverpool can move back above Burnley into sixth place with victory at West Ham United in the late kickoff.
Bournemouth grabbed a last-gasp winner at Newcastle United to move out of the bottom three, above Swansea City who slumped to a home defeat by Brighton and Hove Albion.
Steve Cook sealed Bournemouth's 1-0 win with a 90th-minute header while Brighton's 1-0 win in south Wales came courtesy of Glenn Murray's first-half shot.
Swansea have lost three in a row in the Premier League and were booed off at the end as they fell to 18th place with eight points from their opening 11 matches.
Huddersfield Town beat struggling West Bromwich Albion 1-0 at home to stay safely in mid-table -- Dutchman Rajiv van La Parra deciding it with a curling shot just before halftime.
The hosts hung on in the second half after having defender Christopher Schindler sent off in the 57th minute.
West Brom are without a win in nine Premier League games and, after an impressive opening to the season, are only two points above the relegation zone.
Saturday's early kickoff produced as many goals as the 1500 GMT kickoffs combined with substitute Peter Crouch ensuring a 2-2 draw for Stoke City at home to Leicester City.
Vicente Iborra opened the scoring with his first goal for Leicester before Xherdan Shaqiri levelled.
Riyad Mahrez's fine finish restored Leicester's lead but Crouch headed in Shaqiri's cross for the draw.
(Editing by Clare Fallon)
