Coventry's Cody McDonald slotted home shortly before the hour mark against the run of play on Saturday, but Ross Wallace's equaliser and the late dismissal of Gael Birgimana proved too much for the hosts to overcome.
Austin was on hand in the closing moments to comfortably tuck home and ease the pressure on Clarets boss Eddie Howe.
Andy Thorn handed summer signing McDonald only his second league start of the season, with Howe promoting Keith Treacy to the starting line-up at the expense of Austin.
And the winger, recruited from Preston in July, forced Richard Keogh to block his powerful drive from distance in the ninth minute, before Jay Rodriguez mishit a volley following Wallace's short corner.
The Sky Blues threatened to respond through Lukas Jutkiewicz but his volleyed effort sailed wide of Lee Grant's left-hand upright.
Sammy Clingan then sent a weakly-struck free-kick wide of the target with 23 minutes played as Thorn's men struggled to find a creative outlet.
Their lack of incision nearly saw them punished when Wallace fired into the arms of a grateful Joe Murphy.
And the Clarets had the best chance to open the scoring shortly before the break.
Chris Hussey unconvincingly cleared a Wallace free-kick to concede a corner, the latter whipping in a pinpoint delivery which Andre Amougou nodded agonisingly over Murphy's crossbar.
The visitors came out for the second half eager to atone and looking to add to a solitary goal scored in their previous three outings.
Junior Stanislas found himself in space to drive right-footed at Murphy, who could only parry the ball away, before Austin blasted against a brave Richard Wood when the Coventry stopper was off his goalline.
But it was the 25-year-old former Norwich striker who gave the Sky Blues the lead against the run of play after 59 minutes.
Hussey weaved in behind the Clarets backline to supply McDonald who consummately nodded home his first goal for the club.
Howe's players rallied, Austin smashing against Murphy's body before clipping the rebound wide of the target. Rodriguez arrived in the area following the restart but headed the loose ball tamely into the ex-Scunthorpe man's gloves.
Their reward came in the 73rd minute, Wallace pouncing on Rodriguez' flick-on to hit a sublime strike into the top corner.
Bigirimana saw red in the 80th minute, reducing an already fractious Coventry side to 10 men for the final phases as the visitors continued to press.
And Austin's eighth goal of the season in the final minute proved to be the difference. The former Swindon man arrived in the penalty box four minutes into added time, smashing low into the net to roars of appreciation from the away end.