Half time at London Road sees Burnley head inside with a 1-0 lead over Peterborough United.
Charlie Austin fired the away side ahead inside the first minute, volleying home from an inswinging Ross Wallace cross.
Austin had another effort after 12 minutes, firing a snapshot from 25 yards, though his strike was well saved.
Dean Marney had an effort deflected wide after 22 minutes as Burnley continued to push forward.
Kieran Trippier had another effort on 31 minutes, but his low strike flew wide of the post.