The newcomers had made a dream start to the game with a goal from Charlie Plummer after just three minutes, but that was cancelled out by Daryl Cooper-Smith five minutes before half-time.
Emirates FA Cup
Extra-preliminary round
Winning sides receive £1,500 from The FA prize fund
Sunday 7 August 2016
Tom Neale then gave Epsom the lead with 20 minutes remaining, only for Mark Wallis to draw level with five minutes to go despite Gravesham being down to 10 men.
However the home side finished much the stronger, with goals from Sam Robinson, from the spot, Cooper-Smith and Jordan Gallagher in injury-time to seal the win and a clash with Faversham Town.
Fellow debutants Coventry United earned themselves a replay with their 1-1 draw against Shawbury United but were denied the win by James Askey’s second-half equaliser.
Josh O’Grady had given the home side the lead but they will have to do it all again on Wednesday to reach the next round.
Elsewhere Long Melford were in fine attacking form as they romped to a 5-2 win over St Margaretsbury in an entertaining encounter.
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Scott Yearling scored twice, while Nathan Rowe and Sam Bayliss also got on the scoresheet, along with an own goal, to get through.
AFC Portchester were another team showing their attacking class as they saw off Fawley AFC 3-0.
Steve Ramsey put them a goal up before scores from Tom Froggatt and Joe Bye sealed the victory for the visitors.
Highworth Town also booked their place in the next round after coming from behind to see off Tadley Calleva.
Brett Denham’s goal early in the second half had put Tadley in front, but Highworth responded with scores from Fabian McCarthy and Tom Cole to emerge victorious.
The 2-1 scoreline was a popular one, with AFC Emley, Barkingside and Crawley Green equally victorious over Athersley Recreation, Hadleigh United and Rothwell Corinthians respectively.
Finally Sporting Bengal United and Ipswich Wanderers played out a goalless draw while Halstead Town’s clash with Newmarket Town was abandoned.