Hillians eased out 5-1 winners in the second round clash to set up an away tie at Margate in the next round after a devastating spell just past the half-hour mark.
With both sides resting players ahead of the FA Trophy commitments at the weekend, Ross Morley, Jamie Ampleford, Gary Charman and Tony Nwachukwu, for Horsham, and Hillians’ Pat Harding, Greg Luer and Joe Keehan were all named among the substitutes.
The home side took the lead at a rain-soaked Leylands Park when Horsham were down to ten-man after Byron Napper sustained a heavy gash to the head.
With the midfielder off the pitch receiving treatment, Luc Doherty’s long ball down the right picked out debutant Callum Saunders who shot past Gareth Williams.
Horsham equalised on 24 minutes when Harry Smith and Kieron Tarbie combined to set up Max Miller for a low shot that skidded off the surface and into the net.
Poor defending then allowed the home side to crash three quick-fire goals.
The first came on 33 minutes as the home side lead again when Doherty controlled Dan Perry’s low cross and firing home.
Four minutes later, Perry turned sharply on to Saunders’ pass to hit a first time shot inside the near post, while a minute later Tom Gilbert and Ade Akande gifted the ball to Perry, who stroked home for a 3-1 4-1
In the second half, Gareth Williams saved Horsham’s blushes, as he had done in the first half, by repelling wave after wave of Hillians attacks.
The goalkeeper was, however, at fault for Burgess Hill’s fifth on 52 minutes as a poor goal-kick went straight to Saunders, who controlled the ball before sending it into the net from 30 yards.
Harry Smith registered Horsham’s only shot on target during the half, while the home side forced more saves from Williams and saw an effort cleared off the line by Tom Frankland.
Horsham: Williams, Frankland, Gilbert, Akande, Hills, T.Williams, Napper (Morley 58), Tarbie, Smith (Ampleford), Miller, Bidgood. Unused subs: Nwachukwu, Charman, Hawthorne.