But the Exiles' persistence paid off in dramatic fashion as Boro failed to clear a Josh Sheehan cross and Dolan – who started his career at Middlesbrough – poked the equaliser past Darren Randolph.
It was everything Newport deserved after a bright performance which saw Padraig Amond come close in the first minute and Antoine Semenyo missed two glorious opportunities to pull level.
Newport-born Pulis was proved right to be wary as he named a strong team which included a debut for John Obi Mikel and a rare recall for Stewart Downing despite an ongoing contract wrangle.
Evidently undaunted by the occasion, the visitors – who had seen off Premier League Leicester in the previous round – started well and Amond almost gave them a dream lead inside the first minute.
The visitors continued to give as good as they got although Boro striker Britt Assombalonga came close with a couple of early chances.
Aden Flint headed over from a Lewis Wing free-kick as Boro settled, but Newport looked far from unnerved as Semenyo skied a good chance over the bar on the half-hour.
Wing came closest to breaking the first half deadlock three minutes later, when his free-kick struck the Newport wall and his second effort clattered the top of visiting keeper Nick Townsend's bar.
Driving rain greeted the teams at the start of the second half but Boro's spirit was far from dampened, with Assombalonga coming close again before Ayala put his side in front.
Wing sent in a corner from the Boro left and Ayala rose above the static Exiles rearguard to head home his second goal of the season.
Mikel was withdrawn on the hour to a standing ovation, before Ashley Fletcher came close to extending Boro's lead with a casual-looking finish which was blocked in the box.
Wing came closer for Boro in the 71st minute, drilling a low effort which was well saved by Townsend, only in the team because County's regular number one, Joe Day, is with his wife who is due to give birth to twins.
Townsend produced another fine finger-tip save to stop a deflected effort from George Saville, but despite Boro's creeping domination, Newport would still miss two glorious chances to earn a replay.
Semenyo flashed a free header inches wide from a Dan Butler cross and in the 85th minute and would miss an even better chance before Dolan sparked jubilation among the sizeable visiting support.