Austin, who hit the post with a header in the first half, found the back of the net four minutes after the break.
Half-time substitute Theo Robinson equalised for Derby in the 71st minute but parity was shortlived as Austin scored his fifth goal of the campaign three minutes later.
It was Burnley's first league win in seven matches, with the last coming at Pride Park towards the end of last season.
Striker Jamie Ward recovered from the hamstring complaint that forced him off against Doncaster last weekend to enable Derby manager Nigel Clough to name an unchanged side.
Goalkeeper Lee Grant was dropped against his former club in favour of the long-serving Brian Jensen, who made his first start of the season, as Burnley boss Eddie Howe made three changes from the side that started the Carling Cup win over Barnet in midweek.
David Edgar replaced the injured Michael Duff in defence while captain Chris McCann, who came off the bench to score the extra-time winner against the Bees on Tuesday night, came in for Wade Elliott.
Jensen justified his recall early on with a fine save to keep out a powerful header from Jason Shackell.
Keith Treacy and Ross Wallace sent shots over the crossbar before Burnley's early pressure almost told, but Austin's header struck a post.
Craig Bryson shot wide for Derby before the visitors went close again as Frank Fielding had to save from Wallace.
Burnley took the lead in the 49th minute when Treacy got to the byline and cut the ball back for Austin, who steadied himself with a touch before firing past Fielding from six yards out.
Clough's side came close to an equalising goal on two occasions.
Ben Davies beat Jensen with a 10-yard shot but the ball was cleared off the line by Ben Mee.
Davies was involved again shortly afterwards when his inswinging corner eluded everyone in the box and had to be hacked off the line once more by a Burnley defender.
The Clarets then went straight down the other end and Treacy hit the crossbar with a shot from near the penalty spot on the hour.
Burnley had not kept a clean sheet in the league for 15 matches and some sloppy defending midway through the second half ensured that was extended to 16 games.
Bryson got to the byline after collecting the ball from John Brayford's throw-in and his driven cross was poked home by Robinson from inside the six-yard area.
However, the Rams were not level for long. A long-range shot from Wallace was not properly dealt with by Fielding, who could only push the ball back out to Austin and he bagged his second goal off the game.
Bryson tried his best to get Derby level, forcing a fingertip save from Jensen before going close with two more efforts.
Jay Rodriguez almost added a third goal but his shot was turned round the post by Fielding in the second of six minutes of injury time.