Following a slow start by the hosts, Spurs grew into the game after the opening goal.
Erik Lamela, who returned after a 13-month injury absence in November, scored his first goal since September 2016 shortly after Dan Butler's 26th-minute own goal had given Spurs the lead.
Spurs have not won the competition since 1991 but should Mauricio Pochettino prioritise cup success over finishing in the Premier League top four?
For most of us, playing at Wembley is a dream that will forever be out of reach.
"Of course it was a massive [injury] problem - more than one year out".
And, the result aside, the 7,500 travelling fans of the Welsh side were given a display they can be proud from their team, as Newport pressed high against Spurs and did not let their heads drop after going down.
"Pochettino is from Argentina like me and he knows that it is not easy for a young player to settle in quickly in a completely different culture".
Rigg joined Newport in 2016 and helped the club make a dramatic escape from relegation out of the Football League last season.
The League Two side were magnificent out there.
They are turning into a hardy bunch, finally overcoming Newport in this awkward replay after Dan Butler put through his own net and Erik Lamela scored the second.
Spurs, however, took a fortuitous 26th minute lead as Moussa Sissoko's low cross following Fernando Llorente's square pass struck Newport full-back Butler, with the deflection beating wrong-footed goalkeeperJoe Day at his near post.
Llorente had a couple of half-chances - he smashed a volley wide with the goal gaping - while substituteDele Alli struck the bar with just the keeper to beat.
Tottenham Hotspur now can look ahead to Rochdale away in the next round and trying to navigate their hard playing surface at Spotland.
Sky Sports pundit makes surprise Tottenham prediction
But while it may be tight on the field, Pochettino insisted, off the field this is very far from a meeting of equals. The Reds bounced back from a tough defeat against Swansea City in that game, whereas Tottenham are unbeaten in 2018.
Syrian Army deploys new air defenses in north
The Syria Civil Defence, whose rescue workers are commonly known as the White Helmets, earlier put the death toll at about 30. Two barrels containing chemical gasses had been dropped from helicopters on Sunday night, Saad said.
Fake doctor in India may have infected 21 people with HIV
He said the investigation revealed that Kumar, had used a single syringe for injection in the name of cheaper treatment. In a shocking incident, a quack travelling on bicycle ended up giving HIV to at least 20 persons in UP's Unnao .
No. 5 Ohio State Slips Past Spartans, 4-2
Penn State competed without national champ Jason Nolf (knee) and the Lions also did not have Matt McCutcheon available at 197. The 174-pound bout was a rematch of last year's NCAA final between Penn State's Mark Hall and Bo Jordan of the Buckeyes.
PS4 and Xbox One Fixes
The Xbox One's Patch 18.104.22.168 will address and allegedly resolve "all matchmaking functionality issues" in Monster Hunter World . It's already sold well over five million copies on release and continues to be the talk of most console gamers these days.
Bill Haas shoots 64 to take Phoenix Open lead
He finished fourth in the event last year and was runner-up the year before when he lost in a playoff to Hideki Matsuyama . With the long odds, I like Mickelson as a dark horse option. "It's very helpful and very important to my success ".