Conor Washington netted in either half while centre-back Deji Oshilaja scored his first senior goal for Justin Edinburgh's men.
Despite leading 2-0 in this quarter-final tie, Newport's David Pipe produced a moment of madness in the 67th minute when the Exiles captain was shown a straight red card by referee Graham Scott following a two-footed tackle on Ricky Holmes.
Yet Edinburgh's League Two outfit went from strength to strength and Washington added a third with eight minutes remaining to safely navigate their passage into next month's semi-finals.
Newport were gifted the opener in the fifth minute when Portsmouth centre-back Bondz N'Gala and goalkeeper Trevor Carson got involved in a horrendous defensive mix-up.
N'Gala's attempted header went over the head of Carson and the alert Washington capitalised for his fifth goal of the season to put the ball into an empty net from an acute angle.
Newport doubled their advantage in the 18th minute when Cardiff loanee Oshilaja pounced for the second goal from close range after Robbie Willmott's free-kick was palmed out by Carson.
Pipe gave the visitors a flicker of hope with a shocking challenge on Holmes to receive his marching orders. But Newport relieved the pressure of their numerical disadvantage when Washington's right-footed finish from close-range.