Jacob Bruun Larsen's first-half penalty, accompanied by a Sergio Reguilon red card for bringing down Vitinho in the area, was the early turning point that swung the pendulum in Burnley's favour.
David Datro Fofana then added a second shortly after the hour mark, giving the Clarets a comfortable advantage, as they held out for just a second home win of the season, despite Krisoffer Ajer's header giving them a late scare.
A disappointing defeat for Brentford means it is now six without a win for Thomas Frank's side, who remain 15th due to Nottingham Forest's failure to win at Luton Town.
Burnley also stay 19th, but have closed the gap on Forest in 17th to eight points, with eight games still to play.
The first real moment of note of the afternoon is where everything went wrong for Brentford, as Reguilon was the wrong side of Vitinho and bundled him over in the box, denying a certain goalscoring opportunity.
Referee Darren Bond did not give a penalty on-field, but after a VAR review, he opted to point to the spot and dished out a red card to the Brentford left-back.
Bruun Larsen stepped up with regular taker Jay Rodriguez on the bench, and despite going the right way, Mark Flekken could not keep out the Dane's spot kick, as Burnley took a 10th-minute lead.
Despite the man advantage, Burnley were not entirely dominant in the first half, and they almost gifted the visitors an equaliser, when Dara O'Shea played a blind back pass with goalkeeper Arijanet Muric out of his goal, and the Kosovan had to quickly scurry back and clear the ball on the goalline.
Towards the end of the first half though, Burnley really could have been out of sight, as Fofana missed the easiest of chances from three yards out with an open goal after superb wing play from Lorenz Assignon.
Goalkeeper Muric was promoted to the starting XI due to some underwhelming form from James Trafford, and he justified his selection with a vital save to deny Yoane Wissa just before the break, after some quick thinking from Ivan Toney, who set the DR Congo forward through from a free kick.
At the other end, Muric's opposite number Flekken had his blushes spared, when he dropped the ball to kick up field without realising Fofana was hiding behind him, but the Burnley attacker slipped while nipping the ball away from the goalkeeper, allowing him to recover and clear for a corner.
In the second half, Burnley utilised the man advantage much more wisely, but it was a counter-attack which led to their second goal.
After smart defending from Vitinho denied Keane Lewis-Potter at one end, the Clarets quickly moved the ball up field, and Wilson Odobert's threaded through ball found Fofana, who dinked it beyond Flekken to double the lead.
Odobert was then heavily involved in the next couple of chances for Vincent Kompany's men, being denied by Flekken after good build-up play, before then fluffing his lines when through on goal due to pressure behind him from Vitaly Janelt.
The Bees made it a contest again though with five minutes remaining after a hectic few moments of action, with Muric initially hauling Toney down in the area, but while a VAR penalty check was ongoing, Ajer headed home from Shandon Baptiste's cross, as the Norwegian was the booked for shouting in the referee's face during his celebration, presumably for not giving a penalty on-field.
Brentford then grew into the game, as another Baptiste cross was this time met by Lewis-Potter, but Muric, who had no chance of stopping Ajer's header, kept hold of the substitute's effort.
The hosts sat much deeper looking to hold what they had for the final stages, and it was almost costly, as seven minutes into nine added on, Brentford had a goal disallowed as Toney fouled Muric as Baptiste's cross floated into the top corner.
A late injury to the Kosovo international saw the game go into 103 minutes, but Burnley were able to hold on for a narrow 2-1 success.
Kompany takes his Burnley side to Chelsea after the international break, while Brentford will look to arrest their current winless run against Manchester United at the Gtech Community Stadium.