An own goal from Dan Butler and a well-taken effort from Erik Lamela - his first of the season - ensured that Spurs made light work of their League Two opponents in Mauricio Pochettino's 200th game in charge.
Rochdale will now host Spurs in the next round after seeing off Millwall by a 1-0 scoreline on Tuesday evening.
Newport started the game in North London on the front foot, but the first opportunity of the match went the way of Spurs, with Son Heung-min firing over the crossbar from an acute angle.
The visitors continued to suggest that they were capable of matching their hosts but from the midway point of the first half, it was one-way traffic and soon after Danny Rose had fired into the side-netting with a half-volley, Spurs went in front.
Moussa Sissoko's cross from the right flank was a teasing one but before it reached its intended target, Butler inadvertently deflected the ball inside the near post.
The goal sparked Spurs into life with Lamela seeing a volley from 20 yards well blocked by goalkeeper Joe Day, who then saved from Son's close-range strike.
However, the Premier League outfit doubled their lead with 11 minutes of the first half remaining when Son ran at the Newport defence before teeing up Lamela, who took a touch before slotting the ball into the net from 10 yards.
After the restart, neither side showed enough ambition to suggest that a third goal of the match was around the corner but like in the first half, Spurs gradually found their rhythm.
Fernando Llorente and Lamela found themselves in a two-on-one situation and although it was played perfectly, Lamela's effort from 10 yards was volleyed straight at Day.
Substitute Christian Eriksen sent the ball wide after latching onto a Rose pass before Llorente somehow volleyed wide after meeting a delightful cross from Eriksen.
Pochettino took the opportunity to introduce Dele Alli during the closing stages and the midfielder should have got on the scoresheet after being played through on goal, but the midfielder's rising shot struck the crossbar and went over.
Alli then saw a volley at the back post well saved by Day before Kyle Walker-Peters fizzed a long-range strike marginally wide of the left-hand post.
Padraig Amond should have halved the deficit in added-on time, with Michel Vorm keeping out his 12-yard effort, but it was one of the few moments of panic inside the Spurs penalty area throughout the 90 minutes.
TOTTENHAM HOTSPUR (4-2-3-1): Vorm; Aurier, Alderweireld, Foyth, Rose (Walker-Peters 86'); Wanyama (Alli 78'), Winks; Sissoko, Lamela, Son (Eriksen 61'); Llorente
NEWPORT COUNTY (4-5-1): Day; Pipe (McCoulsky 67'), White (O'Brien 77'), Demetriou, Butler; Tozer (Dolan 59'), Bennett, Labadie, Willmott, Nouble; Amond