Eddie Howe's Clarets suffered defeat in their previous three Championship away games and fell behind once more after Jamie Ward pounced on a loose ball in the 20th minute.
Austin equalised shortly after the half-hour - his eighth goal in as many games and, had Rams forward Conor Sammon displayed Austin's prowess in the penalty area after half-time, victory might have gone to Nigel Clough's side.
But former Swindon striker Austin made it nine for the campaign when he stooped to head Chris McCann's 89th-minute cross past Frank Fielding and give Burnley their first points on the road this term.
Austin recovered from the hip injury that ruled him out of Wednesday night's loss at Leicester to return to the visitors' line-up along with defender Ben Mee and winger Ross Wallace as Joseph Mills, Cameron Stewart and Brian Stock all dropped to the bench.
James O'Connor did likewise for Derby as Jake Buxton came in to partner skipper Richard Keogh at centre-back.
Burnley goalkeeper Lee Grant produced a superb save to thwart his former club when Will Hughes thumped a powerful volley low to his right in the 10th minute.
Grant then kept out Craig Bryson's header from Paul Coutts' cross, but the Rams claimed a 20th-minute lead.
Coutts' corner was only half-cleared and Ward swivelled on the penalty spot to fire home through a forest of legs.
Moments later Grant tipped around the post from Sammon before opposite number Fielding was called into action, denying Martin Paterson on the counter-attack.
But there was little Fielding could do when Austin was left unmarked to smash in McCann's centre after Ross Wallace's cross was blocked.
The momentum suddenly shifted and Burnley came agonisingly close to a half-time lead when Mee thudded a header against the bar from Kieran Trippier's right-wing cross.
Despite the match developing a pleasing end-to-end feel, chances were at a premium early in the second half.
Around the hour Sammon was inches away from finding a decisive touch when Ward swept in an inviting ball from the left.
Former Wigan striker Sammon shot tamely at Grant with 15 minutes to play but the Clarets stopper was more fully extended shortly afterwards, thwarting Bryson from point-blank range.
Burnley were on the rack and Michael Jacobs, on for Hughes, found Sammon to head narrowly over in the 80th minute.
Then, a minute from time, Austin ensured Clough's men paid for their profligacy, and a determined stoppage-time rally could not stop Burnley leapfrogging them in the table.