Subsitute Dack latched onto Adebayo Akinfenwa's header to fire past experienced goalkeeper Billy Turley to earn the League One strugglers a replay.
Town looked on course to knock out their hosts after Joe Martin's 69th-minute own goal.
Despite sitting 57 positions below Gills in the league pyramid, the Northamptonshire side nearly went ahead after just four minutes as leading scorer Steve Diggin hit the crossbar with a volley. Midfielder Steve Towers was then denied by Gills keeper Stuart Nelson after 15 minutes.
At the other end, Curtis McDonald saw his goal-bound shot deflected over by Martin shortly after half-time.
The hosts failed to produce a shot on target in the first half, although in-form striker Danny Kedwell should have done better as he blazed over just before the break.
Town's hero of the last round Turley then kept his side in the game with two key saves to deny Weston and Dack but the latter kept going and fired home in injury time to break Brackley hearts.