Vincent Kompany's Burnley are 19th in the table and eight points adrift of safety, while Brentford are four points above the drop zone in 15th.
Brentford's Sergio Reguilon was shown a red card in the ninth minute -- the fastest sending-off of a player in the league this season -- after a lengthy VAR check determined he grabbed Vitinho as the Brazilian was about to shoot.
Bruun Larsen calmly converted the penalty past the outstretched hands of Mark Flekken, prompting a big celebration from Kompany.
Datro Fofana, who missed a sitter in the first half, doubled Burnley's advantage in the 62nd minute on the counter-attack with a shot into the far corner.
Kristoffer Ajer pulled one back for the visitors with a diving header in the 83rd minute, but the hosts held on for what could be a vital win.
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