- Elijah Adebayo (84')
- Lyle Foster (45'+2')
- Jacob Bruun Larsen (85')
The Danish winger cut inside and fired a curling shot into the far top corner in the 85th minute at Kenilworth Road, a minute and four seconds after Elijah Adebayo equalised for Luton.
South African striker Lyle Foster marked his return from a three-match suspension with Burnley's opener in first half stoppage time.
The goal after a pass from Sander Berge was his third of the campaign.
"It was a very tough game. A good first half from us and then Luton came out and put us under pressure. They got the goal that they wanted but I am just happy we got the win in the end," said Foster.
The victory lifted Burnley, who host Chelsea on Saturday ahead of the international break, above Bournemouth to 18th with four points.
They remain behind Luton and Everton on goal difference after the Hatters won 2-1 at Everton last Saturday.
Bournemouth and bottom side Sheffield United are now the only clubs yet to win in the Premier League.
The match between the two promoted sides was postponed from August due to required improvement work at Luton's aged ground.
It was also the first top flight encounter between the two sides since the 1974-75 season.
The Clarets, who secured promotion as Championship (second tier) winners last season, had taken only one point from their first six matches but had the upper hand for much of the first half.
"We showed what got us through in the Championship last season," Burnley manager Vincent Kompany told the BBC.
"Lyle Foster coming back made a huge difference. The football we played in the first half was really good. We had some good moments. But at the moment we need a few chances to score. Second half was more character."
"I think they shaded the first half, but in the second we completely dominated them, threw everything at them," Luton boss Rob Edwards, whose side host unbeaten Tottenham Hotspur next, told the BBC.
"We performed well tonight and lost which shows the task we have."
- Peter Bankes
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|West Ham United||13||6||2||5||0||20|