Their pressure finally paid off in the 82nd minute when Heung-Min Son expertly used the outside of his boot to flick a corner on to an unmarked Kane at the back post, and this time the England worldwide made no mistake, bringing the score to 1-1 and earning his side another chance at qualifying for the fifth round of the FA Cup.
Newport led for 44 minutes until Kane's 82nd-minute equalizer but the League Two side held on at 10,000-seat Rodney Parade to secure a rich replay at 90,000-seat Wembley Stadium.
Newport's opening goal, unsurprisingly, came from another throw into the box, Ben Tozer launching one from the left which Robbie Willmott hooked to the back post where Amond rose above Kieran Trippier to head home.
With 14 minutes left in the half, Spurs produced their best move of the game, with Kane drawing a fine save from County stopper Joe Day at close range.
Earlier, West Ham manager David Moyes labelled Arthur Masuaku "despicable" after the midfielder was sent off for spitting in a 2-0 defeat by Wigan. "If you do that, then you're going to get the punishment in any walk of life". "I enjoyed it a lot", he said.
Liverpool handed lucky boost before West Brom FA Cup game
James Morrison and Nacer Chadli remain sidelined for the Baggies but other than those two Alan Pardew has a fully fit squad. But, in better news for the Reds, Liverpool captain Jordan Henderson may return after a month out with a hamstring injury.
Stripe Announces Its Is Ending Support For Bitcoin
So far bitcoin is turning out to be more like digital gold than a currency for small or frequent payments. That has made the currency less and less popular with businesses that use Stripe.
Instagram Now Allows Users To Put GIFs In Stories
Typing in "accessories" will bring up GIFs that can be used over faces on photos and videos, like hats, eyes, lips, ears and more. Previously photos/videos would be cropped to fit the frame, but now users can pinch it to share it in its original dimensions.
Wigan have slumped down English football's ladder since winning the FA Cup in 2013, but the League One leaders rekindled memories of that run to Wembley. "It was hard to play forward, always playing safe".
Huffing and puffing for much of the second period, Spurs were finally able to find an equaliser late on through Harry Kane.
Grigg doubled his tally in the 69th with a penalty straight down the middle after Reece Burke was somewhat harshly adjudged to have handled the ball.
Newport, 72 league places below Spurs, made life extremely uncomfortable for their guests from the off. Mauricio Pochettino could have pointed to a poor playing surface at Newport's Rodney Parade ground and a rumbustious atmosphere as reasons his players looked so unnerved from the moment the whistle went - but that would ignore the simple fact that County manager Michael Flynn's team had a brilliant game plan.
Nigeria's Kelechi Iheanacho also scored twice for the Foxes, with Wilfred Ndidi making it 5-1 in stoppage-time. The team didn't really show that they wanted to go through to the next round.