Hartlepool were given a lifeline halfway through the second half, as Andy Sandell received two yellow cards in quick succession, giving Pools the extra man advantage.
Brad Walker scored Hartlepool's first with 10 minutes to go to give the Monkey Hangers hope.
Right-back Michael Duckworth completed the comeback as he found himself in the opposition box, neatly dribbling past several defenders to dispatch the ball into the back of the net.
The result means that Newport drop out of the playoff places to seventh. Pools stay bottom of the League Two table, with manager Paul Murray settling for a draw in his first home game in charge.