Dolan curled home a beautiful free-kick with 13 minutes to go to cancel out Andy Cook’s opener.
And he kept his cool to slot home from 12 yards out in the fifth minute of injury time.
It left Bradford counting the cost of missed opportunities after Cook, Connor Wood and Paudie O’Connor all saw great chances go begging.
Cook missed a great chance to put City ahead when his shot was saved by Nick Townsend on 18 minutes after he took one touch too many.
Dolan fired wide from distance three minutes later, before Danny Rowe hit the wall with a 40 yard free-kick. Nicky Maynard had a shot blocked by Josh Sheehan after a good turn created an opening on 37 minutes.
Cook fired the Bantams ahead after Priestley Farguharson had played him in.
Ward had a free-kick cleared for a corner as Bradford began to step up the pressure, Wood had a long-range effort tipped onto the bar before O’Connor fired wide from the resulting corner.
They paid the price when Dolan gave the Exiles victory at the death after a shot was ruled to have hit a defender’s hand.
Newport County: Townsend, Farquharson, Dolan, Demetriou, Lewis (Ellison 74), Labadie, Bennett (Taylor 58), Sheehan, Shephard, Maynard (Telford 58), Amond (Scrimshaw 74). Unused substitutes: King, Longe-King, Hartigan.
Bradford City: Hornby, A. O’Connor, P. O’Connor, Canavan, Wood, Clarke (Vernam 67), Sutton, Watt, Crankshaw (Stevens 81) Cook (Donaldson 82), Rowe (Evans 67). Unused substitutes: O’Donnell, Cooke, Cousin-Dawson.
Referee: W Finnie (England).
Support The Yorkshire Post and become a subscriber today. Your subscription will help us to continue to bring quality news to the people of Yorkshire. In return, you’ll see fewer ads on site, get free access to our app and receive exclusive members-only offers. Click HERE to subscribe.