Davies was forced into action again to keep out a Duffy header from a corner, before Abdenasser El Khayati came close to producing the opener for The Brewers soon after, as his low shot forced another save.
Duffy was provided an opportunity to score before the break after Beavon nodded the ball down to the striker, but his shot was fired over the bar.
Burton started the second half as the brighter of the two sides, and Phil Edwards twice came close to scoring for the visitors from corners, both times heading over the bar.
However, it was Barnsley who finally broke the deadlock against the run of play, when Josh Scowen was brought down in the box and the hosts were awarded a penalty. Hourihane stepped up to take the kick and fired past Burton keeper Jon McLaughlin.
Duffy came close to equalising for the visitors with a chance from 12 yards, but his effort curled over the bar, before substitute Timmy Thiele clipped the crossbar with a long-range effort.