Dolan, who moved through the youth teams at Boro before leaving for Bradford in 2014, fired home with his first touch to cancel out Daniel Ayala’s 51st-minute opener.
Dolan’s late strike sparked wild celebrations among more than 1,000 travelling supporters, a sizeable contingent being made up of the midfielder’s family and friends.
“It’s unbelievable – you can’t write it,” added Flynn.
“The boy who scored was an ex-Boro player and a Middlesbrough lad, and I don’t think he’s ever scored a goal in the box – it’s a fairytale really.”
Boro boss Tony Pulis paid tribute to Newport’s spirit and persistence but admitted he is not looking forward to returning to the town of his birth for the replay.
Pulis made six changes but still named a strong side with a debut for John Obi Mikel and a rare start for Stewart Downing, and had expected his players to get the job done at the first attempt.
Pulis said: “I thought they (Newport) played with great spirit and togetherness and they deserve all the credit that hopefully they will be given.
“The big disappointment for me was that we didn’t stop crosses – it was almost like we were quite happy for them to stick it in the box and there were lots of occasions we could have got our bodies in the way.”
Pulis paid tribute to Mikel, who played for the first 60 minutes before he left the pitch to a standing ovation.
“It’s the first time he’s had his boots on for a couple of months,” added Pulis.
“John is a quality player, wonderful person, and he’ll be good for us but we’ve got to get him fit. It (the replay) is in between two big games but I’ll get on to Flynny now and get a bucket-full of tickets because I think I’m going to need them.
“We’re still in the cup. In 2011 when I got to a cup final we played Cardiff at home and drew 1-1 and went to Cardiff and won and went on to the cup final.
“Cup games and finals are strange old things.”
Middlesbrough: Randolph, Shotton (McNair 72), Ayala, Flint, Friend, Mikel (Clayton 62), Downing, Wing, Saville, Fletcher (Tavernier 80), Assombalonga. Unused substitutes: Konstantopoulos, Batth, Hugill, van La Parra.
Newport County: Townsend, Poole, O’Brien, Demetriou (Dolan 87), Butler, Willmott, Bennett, Bakinson (Labadie 78), Semenyo, Amond (Sheehan 79), Matt. Unused substitutes: Pipe, Marsh-Brown, Hornby-Forbes, Neufville.
Referee: J Brooks (Leics).