The crowd of 28,565 marked the tournament's highest attendance outside of a final.
After Emily Pierrepont opened the scoring for Barnsley inside 15 minutes, Georgia Gibson equalised with a 30-yard strike midway through the second half. Minutes later, Sharna Wilkinson scored the winner with a long-range effort.
"I'm genuinely lost for words," Wilkinson told BBC Radio Newcastle after the Lady Magpies booked their spot in the third round.
