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.
Horsham boss Dominic Di Paola hails ‘absolutely phenomenal’ Hornets performance as League One Tykes are held in barnstormer
1. Shamir Fenelon of Horsham celebrates his goal to make it 1-1 during the Emirates FA Cup First Round match between Barnsley and Horsham at Oakwell Stadium.
BARNSLEY, ENGLAND - NOVEMBER 03: Shamir Fenelon of Horsham celebrates his goal to make it 1-1 during the Emirates FA Cup First Round match between Barnsley and Horsham at Oakwell Stadium. Photo: Ben Roberts Photo
2. Lewis Carey
Had a nervy start when his clearance was charged down by John McAtee but came back strong to deny John McAtee and could not doing anything to stop Max Watters' point blank header for Barnsley to take the lead. Carey, again, proved pivotal as, shortly after the Hornets made it 1-1, he pulled off a world class save to deny Fabio Jalo. Carey pulled off another spectacular save to deny Max Watters. Carey started off the second half just as well as the first as he pulled of a superb diving stop to keep the Hornets level after the break. Carey was at the ready again towards the last ten minutes with a superb triple save to keep the Hornets in the tie. His amazing display in net helped keep Horsham in the game and without the Hornets could have conceded a lot more than three. 10/10 Photo: John Lines
3. Bobby Price
Price was a rock at the back, holding the Barnsley attack at bay and providing some much needed assurance to the right hand side as well as an attacking threat. 9/10 Photo: John Lines
4. Harvey Sparks
Calmness personified and the heart and soul of the Horsham defence. Sparks was tasked with the enviable task of keeping Fabio Jalo at bay on the left but performed superbly and kept the Portuguese Under 18 international quiet apart from his wonderstrike at half time. 9/10 (Photo by Ben Roberts Photo/Getty Images) Photo: Ben Roberts Photo