Eight-time winners Spurs were just eight minutes away from a humiliating exit when England striker Harry Kane scored to salvage a 1-1 draw away to fourth-tier Newport last month.
But it was a different story at Wembley where, with South Korea’s Son Heung-Min impressing for Tottenham, Dan Butler’s 26th-minute own goal and an Erik Lamela strike eight minutes later put Spurs 2-0 up before half-time.
Tottenham will now travel to third-tier strugglers Rochdale for a last-16 tie on the weekend of February 17-18.
“It’s the FA Cup and it’s difficult,” Tottenham manager Mauricio Pochettino told the BBC. “But I am happy because we wanted to be in the next round.
“Son Heung-Min is doing fantastic this season and he feels important.
“It is important to keep going and winning games. We are going to be very busy in the Premier League, Champions League and the FA Cup but we want to keep this run.”
Meanwhile Newport manager Michael Flynn praised the visitors’ performance.
“It was a good start in the first 25 minutes, but the manner of the goal, the way it went in, it took it out of the boys and deflated them,” he said.
“I am very proud of the players. They put in a huge amount of work and they have done themselves and the club proud.”
But not even the lure of a fixture at Wembley, Spurs’s temporary home while a new ground that will replace White Hart Lane is constructed, could stop Newport midfielder Sean Rigg quitting between the two ties to become a tattoo artist.
The FA Cup has been a huge financial boost for Newport, with their run to this stage earning them at least £700,000 ($972,000, 792,000 euros).
But the concern for the Welsh club was that the chance of an upset had come and gone at their Rodney Parade ground, where a goal from Irish striker Padraig Amond put them on course for a famous giant-killing.
Newport were far from overawed in the replay and they were unlucky when a cross from Moussa Sissoko deflected off Butler and past stranded goalkeeper Joe Day to put Spurs ahead.
It was not long before Spurs had their second goal, when Lamela, getting on the end of a well-weighted through ball from Son, scored for the first time since finding the net against Gillingham in September 2016.
Spurs were comfortable in the second half but late in the game Amond had a chance to add a Wembley goal to his one at Rodney Parade in the original tie only to shoot at Michel Vorm.