Goals from Jack Hunt and Tyreeq Bakinson in the first six minutes of the second half turned the game around for City after Barnsley took the lead in the second minute of the game through Michal Helik.
But in the fifth minute of added time, Michael Sollbauer was fouled and had to receive lengthy treatment before Woodrow was able to take the spot-kick.
The match started at a frantic pace, and within two minutes Barnsley took the lead.
Romal Palmer lofted a corner in towards the back post, and defender Helik was on hand to power home the header from close range to give Barnsley a flying start.
City came close to grabbing an equaliser yet again in the 17th minute when Hunt ventured forward and, with the ball at his feet on the edge of the area, had a strike at goal. The shot deflected upwards and hung in the air for a long time, before Chris Martin met the ball inside the six-yard area, but his header went wide.
City's best chance of the half came five minutes before the break, when a long ball pumped forward by Hunt was headed down by Martin and fell to the feet of Nakhi Wells.
He tried to prod the ball home past an onrushing Jack Walton in the Tykes goal, but defender Jordan Williams was in the right place for his side and was able to clear off the line.
Two minutes after the restart and City were level. A counterattack led by Wells resulted in the striker having two efforts on goal saved by Walton. After a Jamie Paterson shot was then well blocked, Martin laid the ball to Hunt, who had just arrived in the area to rifle an effort past the unfortunate Walton.
A Paterson cross from out wide was palmed away by Walton four minutes later, but his parry fell straight to the feet of Bakinson on the edge of the area, who tried to place his effort and with the aid of a deflection found the back of the net.
After the second City goal, there was very little action for the rest of the half until the 95th minute.
Sollbauer was fouled inside the area by Callum O'Dowda, taking a nasty blow to the head in the process.
After several minutes of treatment and the defender being taken off on a stretcher, Woodrow kept his nerve to convert the penalty.