The Dane's spectacular 85th-minute strike, an early contender for goal of the season, came after Lyle Foster's opener for the visitors had been cancelled out by Hatters striker Elijah Adebayo.
Having recorded their own opening win of the campaign at Everton at the weekend, promoted Luton failed to build on that success, with Rob Edwards' side left to rue missed chances.
"I'm really disappointed with the result; we're down, flat and hurting because of the manner of it," Edwards said.
"We pushed, we had set-pieces, we went forward in numbers and were brave, aggressive and didn't allow them the control they wanted.
"But we come away with nothing and that hurts."
Burnley were second-best for the first 20 minutes, but Vincent Kompany's men rode the storm and began to enjoy some success.
It was the visitors who struck first, in time added on in the first half.
Sander Berge did well to find Foster and he beat Tom Lockyer in a foot race to tuck his effort away.
Luton thought they had done enough to earn at least a point when Adebayo grabbed a deserved equaliser after 84 minutes.
Reece Burke got on the end of a deep cross and provided the knockdown to the striker, who sorted his feet out and tapped home to level.
But their joy was short-lived as Burnley - promoted from the second tier as champions last season - scored their second a minute later to stun a raucous Kenilworth Road into silence.
Bruun Larsen cut inside and produced an outstanding curled effort which found the top corner in front of the away fans.
Luton searched for a second equaliser during seven added minutes but Burnley held on to move up to 18th in the table.
"It was a complete performance in the first half and in the second half it was a battle and we showed character," Burnley boss Kompany said.Â
"Those who know us from the Championship will know we enjoy these type of games as well. It's not something we shy away from."
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