Austin unleashed a stunning 25-yard drive into the top corner three minutes before half-time, ending the Clarets' barren run in front of goal at 348 minutes.
The home side asserted a lot of pressure in the second period but were unable to grab a leveller as they suffered an eighth home defeat of the campaign.
Addicks boss Chris Powell made five changes from the side that lost to Nottingham Forest last weekend. Goalkeeper David Button was handed his first league start for the club and on-loan striker Jon Obika made his full debut, while Lawrie Wilson, Danny Haynes and Ricardo Fuller were brought in.
For the visitors, Alex Kacaniklic was named in the starting XI after completing a loan move from Fulham, while Junior Stanislas, Marvin Bartley, David Edgar and Martin Paterson were all recalled.
Burnley thought they had ended their goal drought after 12 minutes as Junior Stanislas threaded a ball through for Martin Paterson, who put the ball into the net but his celebrations were cut short by an offside flag.
They almost fell behind two minutes later as Johnnie Jackson's corner picked out Danny Haynes in the area but his header was tipped over by Lee Grant.
But it was the Clarets who were on top throughout the first half.
Charlton failed to respond to a quick corner in the 25th minute and Alex Kacaniklic - on his Burnley debut - collected the ball and unleashed a fierce effort that was pushed away by David Button.
Next Stanislas broke dangerously and fed Austin down the left, but the striker was thwarted by a lunging Lawrie Wilson, who successfully deflected the ball wide.
Austin was not to be denied with his next sight of goal as he notched his 25th goal of the campaign in spectacular fashion, leaving Button with no chance from outside the area.
Charlton almost levelled immediately after the half-time interval. A long ball from Michael Morrison was flicked on by Ricardo Fuller towards Bradley Pritchard whose effort was deflected narrowly wide for a corner.
Ten minutes later Button almost handed Austin an easy second as he fumbled Kieran Trippier's cross but he recovered to smother at the striker's feet before the 23-year-old could pounce.
Grant was in more assured form at the other end as he kept his side's lead intact with a brilliant diving save to keep out Danny Haynes' close-range header.
The Addicks finished strongly as they searched for an equaliser. With 10 minutes left Haynes' flick-on found Morrison at the far post but he could not get his header on target and Burnley held on for the win.