The Reds saw a penalty given against them overturned by the video assistant referee midway through the first half which ended goalless after the visitors were the better side.
The hosts were on top after the break but could not break down a resilient away team until Liam Lindsay brought down Matej Vydra in the box in injury-time and substitute Wood converted the spot-kick to cruelly deny Barnsley the replay their performance deserved.
Daniel Stendel made two changes from the side that drew with Luton Town in League One four days earlier. Brad Potts, who had been sold to Preston North End, was replaced by Ryan Hedges with Mike Bahre coming in for Kenny Dougall to play behind striker Cauley Woodrow in a 4-2-3-1.
Barnsley made a very lively start to the match and were the better side in the first 20 minutes without creating any chances. They looked set to go behind when Dimitri Cavare was judged to have brought down Matej Vydra in the box, but the video assistant referee (VAR) system changed the decision as the Czech international was offside from the cross which led to the foul.
Barnsley then enjoyed another decent spell but were reduced mainly to long-range efforts with Mamadou Thiam, twice, and Woodrow missing the target from well outside the box. They had the ball in the back of the net in the 44th minute but Cameron McGeehan’s header from Ben Williams’ cross was rightly ruled offside.
Burnley’s ex-Barnsley loanee Kevin Long had the only effort on target of first 90 minutes when he headed Steven Defour’s corner but his former Oakwell club-mate Adam Davies tipped it over in first half injury-time. The hosts should have taken the lead 12 seconds after the break but Dwight McNeil headed wide from six yards out after meeting Charlie Taylor’s left-wing cross at the back post.
Thiam curled a 25-yard free-kick a foot over the bar then, at the other end, Burnley carved out regular chances as Taylor crosses were headed off-target by Sam Vokes and Vydra whose volley from just inside the box went into the ground and well wide.
Top-scorer Kieffer Moore, who returned to the squad following injury, came on for the quiet Hedges just after the hour-mark as Barnsley changed to 4-4-2.
Burnley dominated the ball in the second half but Barnsley defended well and Davies did not have a save to make until the last-minute spot-kick.
In fact it was the visitors who almost won it late on as Alex Mowatt hit a 25-yard free-kick was just wide of the top right corner in the 87th minute then Woodrow blasted another over from almost the same position soon after.
But it was Burnley’s game as Lindsay slid in on Vydra and this time the penalty decision was not overturned. Wood sent Davies the wrong way to end the Reds’ cup run and put Burnley in the fourth round.
Barnsley team: Davies; Cavare, Pinnock, Lindsay, B Williams; Hedges (Moore 63mins), Mowatt, Bahre (Dougall 83), McGeehan, Thiam (Moncur 77); Woodrow. Subs: Greatorex, Jackson, Adeboyejo, Brown. Yellow cards: B Williams.
Burnley team: Pope; Tarkowski, Long, Gibson, Ward (Gudmundsson 60mins); McNeil, Defour (Cork 77), Hendrick, Taylor; Vydra, Vokes (Wood 74). Subs: Hart, Mee, Barnes, Westwood. Goals: Wood 90+2 (pen).
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