Goals from Joe Nolan and Jacob Holland-Wilkinson capped an impressive performance from the Lilywhites' youngsters.
They face a fourth round visit to either Oxford United or Maidenhead United - those two meet on Friday night.
North End were strong from the off, Tyrhys Dolan seeing a shot saved by Robins goalkeeper Will Buse inside the first minute.
Buse was busy as the half went on, making saves from Brian McManus, Dolan and Holland-Wilkinson.
PNE broke the deadlock in the 44th minute, Chorley-born Nolan volleying home left-footed from the edge of the box.
In the second half, Lewis Coulton went close while Aaron Bennett - on as a substitute - forced Buse into action when his cross caught a wicked deflection and headed goalwards.
North End's clinched victory 11 minutes from time with superb finish on the volley from Holland-Wilkinson.
PNE: Corcoran, Dooley, Williams, Earl, McFayden, Nolan, McManus, Holland-Wilkinson, Dolan (Bennett 71), Coulton, Rodwell-Grant (Potts 84). Subs (not used): Huddart, Lombard, Nicholson, Blanchard, Leigh.
Buse, Towler, C Pearson, Owers, Low, James Taylor, Jamie Taylor, S Pearson, Soady (Francois 46), Conway (Benarous 80), Bell. Subs not used: Smith, Lott, Henry, Gould, Allen.
Referee: Mark Ryder