Newport held Spurs to a 1-1 draw at home 10 days ago but the League Two side were outclassed at Wembley, where a Dan Butler own goal and Lamela's close-range finish sealed a comfortable 2-0 victory.
Tottenham will face League One strugglers Rochdale for a place in the quarterfinals as Mauricio Pochettino's men edged one step closer to ending the club's 10-year wait for a trophy.
Spurs are now 11 games unbeaten since losing so convincingly to Manchester City in December and, with Arsenal to come on Saturday and Juventus next week in the Champions League, would appear to be peaking at just the right time.
Undoubtedly, Newport's best chance of an upset came on the bobbly pitch at Rodney Parade and from the moment Harry Kane equalised there with eight minutes to go, this tie was back in their opponents' hands.
Initially the replay looked awkward for Tottenham, wedged between last weekend's match at Liverpool and the upcoming north London derby, but in fact it allowed several players valuable minutes after recently recovering from injury.
Spurs needed time to absorb Pochettino's 10 changes as a loose pass from Foyth almost set Joss Labadie in on goal but Victor Wanyama traded a yellow card for scything down the midfielder.
The second half followed much the same pattern and the only blot on Tottenham's night was that it failed to yield a much-needed goal for Llorente, who missed a free header from Lamela's cross and then swiped an even better chance wide when picked out by substitute Christian Eriksen.
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- Tottenham Hotspur ease past brave Newport County in FA Cup fourth round replay ...The Independent
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- Erik Lamela ends goal drought as Spurs ease past NewportBuzz.ie
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- Spurs end Newport's Cup run at the second attemptYahoo Sports