The visitors took the lead on 25 minutes through Tony Burnett’s header.
Town were then reduced to 10 men on the stroke of half time as skipper Moses Olaleye was given a straight red card for a challenge on Cole Penny.
Dunstable took off striker Nathan Frater for the second period, replacing him with Newman Carney.
Substitute Lee Roache, back after suspension, missed two glorious chances and Bar made them pay as Lee Close netted a leveller in stoppage time.
Dunstable travel to Ashford Town (Middlesex) on Saturday.
Potters Bar: Mike Underwood, Ashley Harrison-Baker, Lee Close, Ben Bowditch, Tayo Davis-Coker, Jack Friend, Evandro Delgado, Cole Penny, Tom Laxton (Dan Ailey), Josh Hutchinson, James Ewington (Tobi Akinjirrin).
Subs not used: Liam Magwood, Frankie Lane, Sonny Feltwell.
Dunstable: Jamie Head, Jerome Jibodu, Danny Murphy, Joseph Bruce, Marc Leach, Moses Olaleye, Adam Watkins, Tony Burnett, Terry Dixon (Lee Roache 62), Nathan Frater (Newman Carney 46), Christopher Marsh.
Subs not used: Ryan Plowright, Tony Fontanelle, Taishan Griffith.
Referee: Andy Williams.