Wednesday failed to return to winning ways in the Championship, as they went down 3-1 to the Clarets at Turf Moor.
David Jones put Burnley in the lead as early as the seventh minute when he fired a free kick past Wednesday goalkeeper Joe Wildsmith.
Atdhe Nuhiu restored parity for the Owls with a header from Pudil's delicate cross on 20 minutes, but the visitors failed to build on that.
Matthew Taylor fired Burnley back in front on 78 minutes, and late on Andre Gray made it 3-1 to the home team.
Both Bannan and Pudil featured in the game, having joined Wednesday from Crystal Palace and Watford (on loan) respectively in the summer transfer window.
Following the defeat, the duo posted their thoughts on social networking site Twitter.
Good to make my debut not a good result but thought we deserved better and we will be back on Tuesday. Wat a great support thanks a lot— barry bannan (@bazzabannan25) September 12, 2015