Three goals in 10 minutes decided the game, with Tyler Harvey and Conor Hourihane winning it for the Pilgrims.
Chris Zebroski had equalised for Newport but Plymouth regained the lead and held on for the win.
Harvey headed the visitors in front after 20 minutes. The midfielder started off the move by sending a pass out wide to Nathan Thomas, who whipped in an inviting ball for the 18-year-old to nod in his first goal for the Pilgrims.
Zebroski had the away supporters fuming when he equalised just seven minutes later. The ex-Plymouth striker broke down the left side of the penalty area and slotted home, then put his finger to his lips, silencing the travelling fans.
However, they were to have the last laugh when Hourihane put them back in front after half-an-hour. The midfielder fired into the bottom corner from a low cross.