Tottenham, who had needed a late Harry Kane goal to force a 1-1 draw in the first game, were never troubled at Wembley, progressing thanks to an own goal from Dan Butler before Erik Lamela doubled the lead.
There was no place in the Tottenham lineup for top scorer Kane, who was missing along with a host of other first-team players, as Spurs manager Mauricio Pochettino rang the changes with one eye on Saturday's Premier League derby against Arsenal.
The hosts were in cruise control after 26 minutes, however, when Moussa Sissoko's drilled cross deflected cruelly off Butler's heel and wrong-footed keeper Joe Day on its way into the Newport net.
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The lead was doubled eight minutes later when Lamela scored his first Tottenham goal since September, 2016, the Argentine showing quick feet before slipping his finish through the legs of Day having been played in by Son Heung-min.
Tottenham, who missed several chances in the second half with Fernando Llorente, substitute Dele Alli and Lamela all guilty of wasting opportunities, will travel to third-tier Rochdale in the last 16.
(Reporting by Toby Davis, editing by Ed Osmond)
