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.
5. Jack Strange
Strange was a rock at the back, clearing the lines and keeping the Hornets in the tie up against top tier opposition in Barnsley. His partnership with Danny Barker at the back proved pivotal in helping the club secure a famous result. 9/10 Photo: John Lines
6. Danny Barker
Barker in the middle of defence was stellar in the match, sniffing out any danger that came his way and clearing the lines when the Hornets backs were against the wall. His partnership with Jack Strange at the back proved pivotal in helping the club secure a famous result. Photo: John Lines
7. Charlie Hester-Cook
Charlie Hester-Cook battled in midfield and looked composed and threatening whenever he got on the ball. His aggression and tenacity helped the Hornets claw their way back into the game after conceding so early on. He was booked following a robust challenge before being substituted at half time for Lucas Rodrigues. 9/10 Photo: Ben Roberts Photo
8. James Hammond
Hammond bossed the midfield providing assurance and calmness which was personified as he tucked away a penalty to give the Hornets the lead. His experience gave Horsham everything they needed to see out the tie and claim a spectacular draw. 9/10 Photo: John Lines