The result marked an excellent start for Terry Pashley in his first game in temporary charge of the Turf Moor club.
Austin has now scored eight goals in the last five league games and it was enough to beat an under-performing Seasiders side on Saturday evening.
The Clarets had the first sight of goal after Junior Stanislas surged into the box early on, but Stephen Crainey blocked the shot.
Stanislas turned provider after 10 minutes when the League's top scorer, Austin, sprung the offside trap to get on to the pass. He controlled and volleyed towards the far corner but Matt Gilks palmed the ball to safety.
The hosts made their dominance pay eight minutes later, when Ross Wallace produced a wonderful inswinging cross from the right and Austin headed into the roof of the net from close range.
Ian Holloway's men had lost three of their previous four games and struggled to make any inroads with only a wayward shot from Tom Ince to show for their first half efforts, and it was Burnley who continued to threaten.
With three minutes of the first half remaining Gilks was called upon again after Austin's neat flick played in Stanislas - but the Blackpool goalkeeper stood firm to parry for a corner.
Wallace almost capped a lively half with a goal in stoppage-time but could only find the sidenetting with his free-kick.
The Clarets picked up where they left off in the second half, with Gilks needed to save from Austin to keep the deficit at one.
It took almost 10 minutes for Blackpool to register their first attempt of the half and it was Ince who had the opening. Slipped in on the right, he elected to shoot at the near post but only found the sidenetting.
For the first time, Lee Grant was called upon and he produced a remarkable double save on the hour mark. First, Crainey crossed for Alex Baptiste to power a header towards goal and Grant flung himself to the right to tip the ball onto the post. The rebound fell to Gary Taylor-Fletcher with the goal at his mercy but Grant sprung to his feet to parry the ball to safety.
Despite being under more pressure, Burnley always posed a threat and Austin was once again at the heart of it with 15 minutes remaining. He got himself between the two centre-backs and powered a header towards goal from a Kieran Trippier cross - but Gilks was equal to it.
Substitute Sam Vokes almost grabbed a second for the Clarets in injury-time. Gilks thwarted them again but the one goal was enough.