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.
Danny Lafferty returned from injury for the hosts, replacing Joseph Mills at left-back in the only change made by boss Sean Dyche.
Derby manager Nigel Clough also made one change, recalling the fit again Mark O'Brien as James O'Connor dropped back down to the bench.
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.
The visitors replied through Paul Coutts, an old favourite at Burnley's rivals Preston, but Lafferty got in the way of his shot.
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 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.
Austin miscued from close range but Burnley eventually got their second goal when Derby seemingly ignored Duff and he headed in Wallace's corner at the far post.
Substitute Brian Stock worked Legzdins with a long-range effort late on.
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.