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.
9. Jack Brivio
The skipper was solid in his midfield performance. He ran his heart out for the team and gave everything he could to win the midfield battle which he did superbly with his great all-round play. 9/10 Photo: John Lines
10. Lee Harding
Harding was instrumental with his performance against the Tykes. His relentless runs on the right hand side gave the Barnsley full back no time on the ball to settle and dominate play. His defence work on the wing, helping tuck in on crosses from the left as a wing back as well as his own work on the right was superb. His pass to slip in to Ajakaiye for the penalty was also out-of-this-world. 10/10 Photo: Ben Roberts Photo
11. Daniel Ajakaiye
Looked lively starting on the wing and persevered greatly for the first goal, battling hard and wheeling away from two defenders before pulling back to Shamir Fenlon to make it 1-1. Ajakaiye's pace proved dangerous to the Barnsley defence as he nipped in ahead of Mael de Gevigney to win a penalty for the Hornets. He ran his socks off for the team and ran the Barnsley defence ragged before being substituted for Kadell Daniel. 10/10 Photo: Ben Roberts Photo
12. Shamir Fenelon
Fenelon started off really well, playing as the target man, he held up the ball well and bought his fellow attackers in the game and was in the right place at the right time to steer home superbly Daniel Ajakaiye's pull back to make it 1-1. His presence was a constant thorn in the side of the Barnsley defence and his willingness to fight and run for 50/50 balls epitomised Horsham's never say die attitude in the match. 10/10 Photo: John Lines