The Clarets looked to be heading for a crucial victory after Zeki Amdouni broke the deadlock in the 36th minute.
But the Hatters equalised in the second minute of injury time through Carlton Morris' header into an empty net.
His goal left Kompany and the Burnley players and fans enraged after feeling as though goalkeeper James Trafford was fouled in the build-up.
The England youth international collided with Luton striker Elijah Adebayo as he looked to claim Alfie Doughty's cross from the left flank.
He instantly hit the deck with Morris taking full advantage, and despite VAR taking a look, the goal was allowed to stand.
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Despite outrage from the hosts that Adebayo impeded Trafford, former Sunderland left-back Michael Gray insisted it was the right call from the officials.
Speaking on talkSPORT commentary at whether the goal should have been disallowed, he said: "No, I don't. I want to see more of that.
"Because there's physical contact in the game of football, and I just think the way that Trafford came out of his six-yard area, he was the one searching to try and clear is lines.
"Adebayo had nowhere to go, he was static, he [Trafford] just needed to make contact which is exactly what he did.
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"And when it fell luckily for Carlton Morris, he did what centre-forwards do, and that's put the ball in the back of the net."
The result leaves both clubs in the relegation zone, though 19th-placed Burnley trail Luton by four points.
Luton are now level with Everton in 17th, however they are in the bottom three due on goal difference.