Burnley came close again just moments later, this time Vokes providing a low cross to Austin, but he was denied twice at the near post by Tom Heaton.
The Robins keeper was again called into action just a few minutes after, when he stopped Austin's header following a good Ross Wallace cross.
Dean Marney then had a go from 30 yards out for the hosts, but his effort curled fractions wide of the post.
Burnley continued their early onslaught and Austin was inches away from giving them the lead in the 19th minute, after spinning away from the last defender before firing across goal but just over the angle.
Despite the home side's dominance, it was Bristol who took the lead 10 minutes from half time, when Steve Davies's through-ball led to Adomah rounding goalkeeper Lee Grant before rolling the ball into the back of the net.
Burnley nearly found an instant response, but Austin's bullet header from Wallace's corner flew just over the bar.