Two first-half Gary Hooper goals saw the Reds behind at the interval before Jack Stobbs added a third late in the second-half following the dismissal of Reds defender Cole Kpekawa.
Kpekawa started alongside Adam Jackson in central defence with Callum Evans and Louis Wardle in the full-back positions; Jared Bird shared central midfield with George Moncur who was in action for the first time since returning from a loan-spell with Peterborough United. Out wide were Omari Patrick and James Carvell, with Elliot Lee partnering Jacob Brown up front.
The visitors took an early lead on seven minutes when after good work by Stobbs down the right, the Reds defence failed to clear their lines and Hooper had the simplest of tap-ins for 1-0.
The Reds grew into the fixture afterwards with Patrick going close at the far post and a rasping drive from Brown whistled just over the Wednesday bar. On 20 minutes Patrick was slipped in down the right by Brown and his cut-back from the byline was begging to be slotted home but Carvell couldn't quite stretch enough to get there.
Wednesday went down the other end again with Stobbs and his poor cross was missed by Jackson attempting to clear and there was Hooper to tap home for 2-0.
The away side didn't get out of their half for the rest of the first-half but the Reds, dominant in possession and territory couldn't quite muster up any clear opportunities until Brown was found inside the box but denied by Dawson in the visitors net with a good save from the snap-shot.
HALF TIME: Barnsley 0-3 Sheff Wed
Having created one or two half-chances at the start of the second-half, the Reds carved out a real opportunity when Moncur found himself just inside the box to the left and struck a left-footed effort towards the top corner only for Dawson to fling himself across to repel the shot.
Almost instantly, Barnsley were reduced to 10 men when Kpekawa went in high with a tackle on a Wednesday midfielder. That saw Harsley make his first change as he brought on Julio Rodriguez to shore up the back-line, sacrificing Carvell.
Despite the reduction in numbers the Reds had a real good go at the opposition with Moncur becoming increasingly influential, driving his team forwards.
Jake Charles and Matty Wolfe replaced Lee and Wardle with 15 minutes remaining.
On 82 minutes another stinging Moncur shot tested the Owls 'keeper from just outside the box before Julio's lovely run and cross from the left saw the sliding Brown just failing to get a touch on it to steer it home.
Wednesday broke quickly from the restart and following good play by the impressive Sean Clare, it was Stobbs who rounded off the 3-0 victory when he placed a shot beyond Nick Townsend from 10 yards out.
FULL TIME: Barnsley 0-3 Sheff Wed
Reds XI: Townsend; Wardle (Wolfe 77), Jackson, Kpekawa, Evans; Patrick, Moncur, Bird, Carvell (Julio 58); Brown, Lee (Charles 77).
Unused subs: Walton, Dolan.
We will bring you further reaction to today's fixture on Barnsley Player HD.
The Under 23s are next in action this coming Sunday afternoon away at Huddersfield Town.