After going a goal behind early on, Horsham equalised through Shamir Fenelon’s excellent finish before James Hammond gave the Hornets the lead from the spot.
Barnsley overturned the deficit to make it 3-2 before Richards tapped home late on to seal a famous cup result.
But who stood out for Horsham? Here are Sam Pole's ratings from the game.
13. Lucas Rodrigues
Lucas Rodrigues came on and gave a superb performance in midfield and was pivotal in Horsham equalising late on. Luca Connell and John McAtee were both denied in quick succession before Rodrigues broke away to square it to Tom Richards to tap into an empty net. 9/10 Photo: John Lines
14. Kadell Daniel
Daniel came on and provided a superb burst of energy to Horsham's front line, his tricky work on the wing as well as his pace kept the Barnsley players on their heels throughout the second half. His constant running and superb play helped keep the Hornets in the tie. 9/10 Photo: Ben Roberts Photo
15. Doug Tuck
After coming on Tuck provided extra calmness into the midfield to help keep the Hornets in control of the midfield. His defence work late on gave Horsham the foundation they needed to stay in the tie and seize a memorable cup performance. 9/10 Photo: John Lines
16. Reece Myles-Meekums
Myles-Meekums came on and did what Fenelon had started at the top of the pitch. His superb work and pressing helped Horsham stay in the game and he worked his socks off to hurry and harass the Barnsley back four. His link up play, despite the visitors' backs being up against the wall late on, was superb as well. Photo: John Lines