Ashley Barnes scored his third league goal of the season from the spot with 12 minutes remaining after Gabriel Osho handled substitute Vitinho’s flick inside the penalty area.
The Hatters were reduced to 10 men just seconds after as skipper Tom Lockyer was shown a red card for dissent by referee Jeremy Simpson.
It was a fifth away win in succession for the Clarets, a 12th game unbeaten in the Championship, a 13th clean sheet and a 29th consecutive game in which they’d scored.
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It was a fifth away win in succession for the Clarets, a 12th game unbeaten in the Championship, a 13th clean sheet and a 29th consecutive game in which they’d scored.