The Fulham loanee, who had opened his account for the Reds in the FA Cup, struck in the 36th minute against their fellow promotion-chasers in front of a ground record attendance of 4,801.
But left-back Zeki Fryers was dismissed in the 64th minute for a second yellow card and Woodrow was sacrificed as defender Dani Pinillos came on.
The Reds kept their nerve and took the points with Kieffer Moore tapping into an empty net in the fourth of the six minutes of stoppage time.
Emergency loan signing Nottingham Forest goalkeeper Jordan Smith made his debut for the Reds at the Crown Ground.
A posterior cruciate ligament injury sustained by second-choice Jack Walton in the midweek Checkatrade Trophy win over Bradford City ruled him out of this Roses game and with regular No 1 Adam Davies on international duty with Wales, Barnsley were forced to act quickly.
Smith was the only change to the side which overcame Notts County in the FA Cup.
Stanley made two changes from the previous Saturday with striker Offrande Zanzala returning from injury and Nick Anderton replacing Ben Richards-Everton in defence.
Barnsley product Jordan Clark and Sean McConville forced fine saves from Smith with rasping drives in the early stages as Stanley drove down to the open end packed with the 2,352 Reds fans.
Barnsley hit back with Brad Potts racing clear from halfway and feeding the supporting Dimitri Cavare, whose shot was beaten back into his path and his ensuing cross was smothered by Connor Ripley.
Barnsley struck after more pressure from Stanley. Moore received the ball down the right and a clever back-heel enabled Woodrow to race in and sidefoot home in the 36th minute.
Moore almost made it 2-0 on the stroke of half-time with a shot on the turn which went just too high and wide.A goalbound volley from Potts shuddered off the head of central defender Mark Hughes immediately on the resumption and Mamadou Thiam forced a good save as he cut in from the left.
In reply, McConville’s curling free-kick from the left floated just wide of the far post.
Thiam curled a shot narrowly over after Alex Mowatt’s corner had been headed out but it was Smith to the rescue at the other end as pawed away a deflected cross from McConville.
Fryers, however, received his second yellow card for a foul on Callum Johnson near halfway in the 63rd minute to leave the Reds down to 10 men.
However, Barnsley created the better chances in the closing stages and Moore finished the job when Ripley could only parry a drive from Potts up and behind him, allowing an unmarked Moore to knock the ball in.