After a bright start from the visitors, it was Leeds who ended up having the better of the chances in the opening 45 minutes, but the first half finished with no goals for either side.
The hosts had a chance early in the second period after Jason Shackell fouled El Hadji Diouf to give away a free kick in a dangerous position. However, Lee Grant was able to stop Luke Varney's shot following Ross McCormack's delivery.
Leeds's dominance paid off in in the 62nd minute when Diouf fed the ball into Austin, who kept his shot low to beat Grant in the Forest goal.
Stephen Warnock looked to double the hosts' advantage five minutes later, but Grant came off his line to collect the ball before his ross connected with a teammate.
Burnley created few changes in the second half, with Paddy Kenny rarely troubled in the Leeds goal.
The result leaves Leeds 11th in the Championship table, while Burnley slip to 17th.