Eddie Howe's side displayed the kind of form that marked them out as top-six contenders at the turn of the year as they had the better of an entertainingly open first half.
The decisive moment came after 27 minutes when Austin coolly converted Josh McQuoid's cross at the second attempt.
Brighton stepped up their efforts in the second period, but a Burnley defence superbly marshalled by Michael Duff stood firm.
The Seagulls were reduced to nine men in a bad tempered defeat to the Clarets in December and tensions came close to spilling over in the closing stages as Howe's men secured a victory that moves them up to 13th place.
Gus Poyet's side have now taken two points from their last three outings, an untimely slump that will see them replaced in the play-off spots by Middlesbrough or Cardiff depending on the result when both sides meet at the Riverside Stadium tomorrow.
Burnley winger Junior Stanislas was rewarded with a starting place following his impressive cameo during Tuesday night's 3-1 defeat to Birmingham.
Gary Dicker made his first appearance since suffering a broken leg in October as one of three changes made by Poyet.
Gonzalo Jara returned from a three-match ban to join Dicker in midfield, while Craig Mackail-Smith was recalled to partner former Burnley loanee Sam Vokes up front.
After a pair of dangerous early crosses from makeshift Clarets left-back Ross Wallace caused panic in the Brighton box, the visitors settled into a controlled period of possession with former Valencia playmaker Vicente to the fore.
Dicker was given a robust welcome back to action by Burnley midfielder Marvin Bartley, who escaped with a booking after a nasty late challenge.
Bartley ultimately came off second best and hobbled out of the action to make way for Dean Marney with 31 minutes gone, by which time his side were in front.
McQuoid made the most of the space afforded out wide by Brighton's narrow midfield four and his right wing cross picked out Austin, who calmly rounded Seagulls goalkeeper Peter Brezovan to finish after Gordon Greer blocked his initial first-time shot.
Burnley enjoyed further joy down the Brighton left and Danny Ings headed a teasing delivery from Kieran Trippier wide before Austin almost capitalised when another 37th minute centre from the former Manchester City youngster caused further havoc in the Seagulls area.
Grant plunged to his left to make a superb save when Vokes teed up Mackail-Smith two minutes after the restart.
Vicente failed to capitalise on being hacked down by Duff in the 55th minute, scuffing his 25-yard free-kick into the bottom of the wall.
Moments later the Spaniard again failed to make clean contact and dragged Mackail-Smith's lay-off past the far post.
Ashley Barnes replaced Vokes after an hour and his first contribution was to blaze over from Joe Mattock's low left wing cross.
Austin speared a shot narrowly wide from McQuoid's 77th minute knockdown as the Clarets gained some well-earned respite.
Poyet appeared to feel his side were on the wrong end of many decisions, but his players hardly covered themselves in glory as Vicente and substitute Craig Noone collected petulant late bookings.