Owen Coyle is still waiting for his first home win as Burnley manager after Tamas Vasko netted for Bristol City.
Gabor Kiraly denied David Noble, while Darren Byfield blazed wide and had a header saved as City began brightly.
Ade Akinbiyi hit the post just before the break for Burnley and Adriano Basso produced a superb save from Jon Harley.
Vasko netted from Michael McIndoe's free-kick to make it 1-0 but Akinbiyi again hit the post and City hung on despite coming under late pressure.
Burnley boss Owen Coyle:
"We are just not being clinical. We had seven or eight real goalscoring opportunities but it's all about composure and sometimes a bit of luck.
"But you have to stay strong, I would be more worried if we weren't creating chances.
"If the players continue to give that effort and keep persevering they will get their reward but we can be more clinical with our finishing."
Bristol City manager Gary Johnson on Vasko:
"I watched a video of him playing and he beat up a centre forward for 90 minutes.
"He has good feet, he's a big lad and we invited him across. He has been waiting for his opportunity. Tamas is decent in the opponents' box and he took his chance well.
"He has waited for his place and has been sharp in the reserves. He proved that we needed to be a bit more solid."
Burnley: Kiraly, Carlisle (Jones 33), Caldwell, Unsworth, Jordan (Rodriguez 86), Elliott, James O'Connor, Harley, Lafferty, Akinbiyi, Blake.
Subs Not Used: Jensen, Garreth O'Connor, Kay.
Bristol City: Basso, Orr, McCombe, Vasko, McAllister, McIndoe, Noble (Brian Wilson 80), Elliott, Sproule, Johnson, Byfield (Trundle 61).
Subs Not Used: Weale, Showunmi, Ribeiro.
Goals: Vasko 67.
Ref: Clive Oliver (Northumberland).