Burnley came close again just moments later, this time Vokes provided a low cross to Austin, who was denied twice at the near post by Tom Heaton.
The Robins goalkeeper was again called into action just a few minutes after when he brilliant 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 City who took the lead 10 minutes from half time, when 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.
It took the hosts just seven minutes into the second half to find an equaliser, and they got it when Shackell nodded in Junior Stanislas's bouncing corner.
Burnley nearly went in front just five minutes later, but Heaton saved Vokes's poor header from a Wallace cross before McCann hit a half-volley straight at the keeper a minute after.
However, Burnley completed the comeback in the 61st minute, when Kieran Trippier's cross found McCann at the back post and he took a touch before smashing the ball past Heaton to make it 2-1.
Bristol were given the perfect opportunity to find an equaliser of their own eight minutes from the end when they were awarded a penalty after Adomah was brought down by Joseph Mills.
However, Davies's spot kick was magnificently save by Grant to keep Burnley in front.
Burnley nearly extended their lead with four minutes to go, but Heaton turned Austin's 25-yard effort brilliantly past the post.
The home side did find a third goal in stoppage time when Paterson's initial header hit Matthew Bates before falling kindly to the striker, who was able to volley past Heaton to secure the victory.