The League Two side had appeared on course for a huge shock when Padraig Amond broke the deadlock after 38 minutes, heading home Scot Bennett's cross.
Spurs' pressure towards the end became increasingly intense and, although Newport often found themselves with 11 men behind the ball, their desire failed to trump the visitors' desperation, as Kane tapped in to give the away side the opportunity to do it all over again at Wembley.
As a result, Newport will head to Wembley Stadium for a replay in the week commencing February 5.
Now Pochettino's side are facing a replay to add to an already hectic fixture schedule, with a two-legged Champions League tie with Juventus also on the horizon.
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Do you know anybody?" "I'd like to meet somebody who can swing it over 120 miles an hour with a fused back", he said. Open victory, though won't be among the favourites this week after a long lay-off with injury.
West Ham were without several injured first-choice players and an upset always looked on the cards. The England striker scored from a corner in the 88th minute.
Two minutes later, a long throw from Ben Tozer was launched into the penalty area.
Labadie came off and was replaced by Matt Dolan, with the home side holding on for a draw, Kane firing narrowly wide in stoppage time, as Spurs pushed hard for a late victor.
Hugo Lloris, Toby Alderweireld, Harry Winks and Danny Rose were also missing from the squad, meaning rare starts for Kyle Walker-Peters, Juan Foyth and Fernando Llorente.
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With 32 minutes gone Kane was within a whisker of giving the Londoners the lead, hitting the post after combination play between Llorente and Moussa Sissoko had created the opening. In the 67th minute, Heung-Min Son nearly made it 1-1. "But the team didn't show really that they want to go to the next stage of the FA Cup". It was half cleared as it fell to Robbie Willmott on the corner of the box.
It was a rare moment of concern for Newport but Rodney Parade erupted six minutes later.
The 27-year-old was the standout performer in the first half with his neat passing and tough tackling, only for a 32nd-minute booking to put a brake on his excellent display.
"I'm a little bit relieved - well a lot", he told reporters at his post-match presser. "We've got good people running this club and I'm sure they'll make the right decisions". But I've got every confidence in my players because they've been magnificent from day one.
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"I think that we were aware that the key to playing today was having the capacity to challenge them. We can not use the pitch as an excuse, we need to look at us".