Cheltenham Town 0 Burnley 1
In front of Steve Cotterill, former manager of both clubs, Cheltenham won two early corners, but Burnley had the first shot at goal as Kieran Trippier picked up a Ross Wallace pass and fired a left-foot effort wide.
And Wallace then forced a save from Scott Brown moments later, turning the ball round the post for the first of two corners, the second of which broke to David Edgar, who leaned back and sent a great chance over from 12 yards out.
Austin had a header deflected over from a Junior Stanislas corner, and from Wallace's subsequent flag kick, Dean Marney dummied and Trippier dragged a shot wide.
At the other end, Russell Penn skidded a shot past the post, although there was an off-side flag raised, but Burnley continued to dominate, as Wallace sent a fierce strike just wide from 30 yards.
Wallace then had a shot saved from close range after a Joseph Mills throw was chested on by Austin for Chris McCann, who swept the ball to the right of the box.
A weak back header from Edgar almost let in Alan Bennett, but while he lifted the ball over Lee Grant, it went across goal with no-one following up.
Just after the half hour, the Clarets got the goal they deserved as Wallace took a short corner with Stanislas, whose cross picked out Austin at the back post for a thumping header back across the keeper.
Martin Paterson's measured curler was somehow tipped against the angle and behind by Brown, and Stanislas whipped a free kick wide towards the end of the half, before Austin had a header turned round the post from a Wallace cross, and Joseph Mills bent a righ-footer just wide from Paterson's lay-off.
Paterson then hit the upright from McCann's pass after sneaking the ball past Brown, as the Clarets went in 1-0 up.
Burnley: Grant; Trippier, Edgar, Shackell, Mills; Wallace, Marney, McCann, Stanislas; Paterson, Austin. Subs: Lafferty, Duff, Vokes, Ings, O'Neill, Porter, MacDonald, Long, Hewitt, Mee, Stewart.