Elliott opened the scoring on 81 minutes, hooking home a loose ball for his first goal of the season.
Myles-Meekums then ensured the Hornets’ passage into the last 32 three minutes later, tapping in Greg Luer’s cross from close-range.
The FA Trophy fourth round proper draw will be held tomorrow (Monday, December 15) at approximately 5pm.
The draw will be hosted on talkSPORT 2 by Olly Clink, with Adrian Clarke drawing the balls.
Fourth round matches will be played on Saturday, January 10, 2026.
Horsham, now unbeaten in 17 in all competitions, are back in league action this Saturday at home to Chesham United.
See pictures by John Lines from Hemel Hempstead v Horsham on this page and those linked – and get the latest from Dominic Di Paola in the West Sussex County Times, out on Thursday.
13. FA Trophy: Horsham FC strike late to see off Hemel Hempstead - the match in pictures
Late goals from Jake Elliott and Reece Myles-Meekums at Hemel Hempstead Town saw Horsham FC advance to the fourth round of the FA Trophy for just the second time in the club’s history. Photo: John Lines
14. FA Trophy: Horsham FC strike late to see off Hemel Hempstead - the match in pictures
Late goals from Jake Elliott and Reece Myles-Meekums at Hemel Hempstead Town saw Horsham FC advance to the fourth round of the FA Trophy for just the second time in the club’s history. Photo: John Lines
15. FA Trophy: Horsham FC strike late to see off Hemel Hempstead - the match in pictures
Late goals from Jake Elliott and Reece Myles-Meekums at Hemel Hempstead Town saw Horsham FC advance to the fourth round of the FA Trophy for just the second time in the club’s history. Photo: John Lines
16. FA Trophy: Horsham FC strike late to see off Hemel Hempstead - the match in pictures
Late goals from Jake Elliott and Reece Myles-Meekums at Hemel Hempstead Town saw Horsham FC advance to the fourth round of the FA Trophy for just the second time in the club’s history. Photo: John Lines