A goal looked a certain outcome but Potterill came from nowhere, from Jones' left, to make a saving block after Jones pulled the trigger from inside the penalty area.
Crowd pleaser Leroy Phillips screwed a shot wide for the home side after 58 minutes and was pushed over by Harwood three minutes later by the left corner flag and the Bromley shot-stopper was shown a yellow card.
Reid did well to surge forty yards with the ball at his feet and his run was ended by Jay Westwood who chopped down the powerhouse striker. This resulted in a small brawl, which saw Westwood booked for the initial challenge and Luckett booked for his part in the aftermath.
Beale's left footed effort from 22 yards sailed high and wide of the left post after 65 minutes.
Bromley were at this stage confident and took the game to the home side in the final twenty minutes and created a couple of chances to win the game.
Ashford were forced to play the closing exchanges with only ten men as Jones was taken off the pitch due to injury.
Drewett hooked a shot over from Luckett's high cross before a move involving Drewett and Huggins saw Reid powerfully shoot wide of the left upright.
Beale drilled a low shot narrowly wide of the left upright, but Ashford's unbeaten run to the season nearly came to an end in the 88th minute.
Drewett surged with the ball and fed Potterill after he made an overlapping run. His cross found Adolph Amoako, who replaced the hardworking Reid in the 84th minute, and the former Crystal Palace striker leapt and headed towards the top right hand corner, but Westwood headed the ball off the line to deny Bromley victory.
Bromley captain, the solid central defender, Dean Forbes said: "A much better performance than the league games last week when we rolled over and died against Walton & Hersham last Saturday. We should have never lost at Windsor & Eton in midweek and today we were never going to get beat."
The replay will take place at Bromley's Hayes Lane ground on Tuesday, 16 September, 7.45pm, and Forbes added: "Ashford will get exactly the same treatment in the replay as they got in the second half today. They will get the ball straight down their throats and it's just a matter of time when Tony Reid and Leroy Huggins gets us another goal."
Forbes praised Bromley's front two, saying: "Reid and Huggins today led the line well and took a lot of pressure of the back four. When the ball goes upfront the ball is sticking and it's good for the midfield because we have got two busy people there."
Ashford Town: John Whitehouse, Martin Anderson, Aaron O’Leary, Ian Ross (Capt), Rob Gillman, Lyndon Tucker (Jay Westwood HT), Michael Beale, Tom Adlington, Paul Jones, Lee McRobert (Adrian Stone HT) Lewis Phillips (Barry Gardner 72) Subs: James Gardner, Kevin Skinner.
Bromley: Danny Harwood, Steve Potterill, Colin Luckett, Joe Vines, Mark Willy, Dean Forbes (Capt), Gary Drewett, Danny Smith, Tony Reid (Adolph Amoako 84), Leroy Huggins, Jason McKoy. Subs: Andy Hunt, Bobby Nolan, John Butler, James Goodwin
Referee: Mr R Briffitt
Assistants: A Fissenden and J Bradley