The Clarets looked ready to make an exit from the bottom three, until Tony walked into Christian Eriksson's lovely cross for four minutes.
Tony won a penalty deep in injury time, Nathan Collins sent off to bring him down, and he slammed in a spot-kick to seal the win.
Burnley are one point behind 17th-placed Everton, who have two games in hand, while Brentford are now nine points away from the Clarets.
Burnley had the best chance of the game until a late attack when Jay Rodriguez's 30-yard strike was cannoned from the crossbar.
He had only one attempt on target, when a formidable Rico Henry backpass was played into the Maxwell cornet but his shot was blocked by David Raya.
Tony gives a big blow to Burnley
Burnley always seem to be out of trouble with relegation despite a poor start but his six-year stint in the top flight has rarely been in such danger.
Back-to-back victories in February - their only win of the season along with a win over Brentford in October - have been a one-point chase from four games.
Sean Dyche's side either need to win more games in the last 11 games of the season than the first 27 - or else hope Leeds and Everton, who have recently changed managers, continue to lose.
He didn't create enough chances to win, with Cornet's tame finish and Rodriguez's rumblings having equal moments of chance.
To make matters worse, Collins' late red card to deny Tony a clear goal-scoring opportunity reduced his options with captain Ben Mee still injured.
Meanwhile, Tony alone is pulling Brentford to safety - with five goals in the past eight days against teams in the relegation zone.
His hat-trick took Norwich into a 3-1 win last weekend and now has eight goals in his last six games.
Thomas Frank's side has 30 points and this season's teams do not need to reach 38-points to stay afloat.
They were far from brilliant but they had the best chances before Tony's late double.
Ericsson, making his first home debut, had saved a 20-yard deflection shot by Nick Pope and Vitaly Genault leading the bar.