Austin struck five minutes before half-time. The former Swindon striker was at his predatory best to turn the ball into the net after the visitors failed to deal with Kieran Trippier's cross.
Some more poor defending cost the Rams after 74 minutes when Michael Duff headed in the second goal.
In a game of few chances, it was enough to see Burnley maintain their recent stranglehold over Derby. The Clarets are now unbeaten in eight games against the Rams, winning six.
Rams goalkeeper Adam Legzdins saved early on from Martin Paterson before Chris McCann and Ross Wallace had efforts blocked by Richard Keogh and Conor Sammon respectively.
Dean Marney headed weakly wide just after the half hour mark from a good far-post position before Austin struck. Trippier's cross from the right-hand side was not properly cleared and Austin swept a low shot past Legzdins from 12 yards out.
Derby might have equalised before the break but Mark O'Brien headed wide from Michael Jacobs' free-kick.
O'Brien made a block to deny Austin another goal early in the second half and Wallace's shot from the right side of the penalty area hit a post just before the hour.
Burnley got their second goal when Derby seemingly ignored Duff and he headed in Wallace's corner at the far post.
Derby substitute Theo Robinson had the last chance of the game but missed the target from 12 yards out as the Rams suffered a seventh away loss in the league this season.
- Sports & Recreation
- Ross Wallace
- Kieran Trippier
- Adam Legzdins