The draw leaves Spurs - who play Manchester United, Liverpool and Arsenal in the next three league games, as well as facing Juventus in a two-legged Champions League round-of-16 tie - facing an unwanted replay at Wembley.
Wanyama blasted wildly over from distance and Spurs made their final change, bringing on Ben Davies and taking off Mousa Dembele, and Davies soon fired in a unsafe cross that was scrambled behind for a corner after 82 minutes.
Two minutes later, a long throw from Ben Tozer was launched into the penalty area.
With Spurs struggling going forward, there was no riposte forthcoming before the break as Pochettino's men trailed at the interval.
Tottenham: Vorm 6, Trippier 6, Foyth 6, Vertonghen 6, Walker-Peters 6, Dier 5, Wanyama 5, Sissoko 5, Dembele 5, Llorente 4, Kane 6.
In the 68th minute, a flowing move saw the ball make its way to substitute Heung-min Son, but his effort was well saved by Day.
But Day was beaten in the 82nd minute as Newport's defences were finally breached. But their hopes of a historic victory were dashed when Harry Kane put one in for the visitors just eight minutes before full time. The team didn't really show that they wanted to go through to the next round.
Newport avoided defeat against Spurs for the first time in their history, having previously lost all four meetings.
82' - GOAL! Newport 1-1 Tottenham (Kane): Spurs have the equaliser, and who else but Harry Kane who nets his 30th goal of the season.
Kane has scored in four consecutive games for Spurs in all competitions, his best run since November 2016.
"I hear in five years here that the FA Cup is magic". It was half cleared as it fell to Robbie Willmott on the corner of the box.
It seemed Rodney Parade, where Newport Rugby Club beat giants New Zealand and South Africa in the 1960s, was about to play host to another great sporting upset.
"If the replay is anything like the second half then I'm not relishing it".
"I'm gutted if I'm honest because we conceded so late in the game", he told reporters. "We will see what happens in the last few days".
His worldwide fellow, Lucas Moura, is also on the list of potential candidates, as the player stressed his desire to wear the white shirt. But I've got every confidence in my players because they've been magnificent from day one. "We need to take the individual responsibility and understand that, in football, we need to show more".