Exeter got off to a flying start at Rodney Parade, taking the lead in the 22nd minute when Tom Nichols converted a penalty after being brought down by Darren Jones.
Newport almost found an immediate response, only for Chris Zebroski to rattle the crossbar.
That miss looked more costly in the 40th minute when Graham Cummins coolly curled home to make it 2-0.
Yet County showed impressive character to fight back, pulling a goal back in the 49th minute as Yan Klukowski converted from Ryan Jackson's long throw.
And the hosts drew level two minutes later, Max Porter powering home from distance to cap a remarkable turnaround.
Both sides had chances to win it, yet a decisive fifth goal was not forthcoming as the spoils were shared.