The League Two side led for half of the game but will have to settle for a lucrative replay at Wembley after Kane cancelled out Padraig Amond's header.
Newport showed straight away that they were not going to be intimidated by their esteemed visitors and should have taken the lead on three minutes.
It took until the 31st minute for Tottenham to register an attempt on goal, and Kane came within inches of putting them in front when he shot against the post from Moussa Sissoko's cutback.
Spurs were spooked and a scuffed clearance from Dier presented Amond with a chance for a second, with Vorm tipping his shot from a very tight angle onto the woodwork.
The visitors marginally improved in the early stages of the second half but missed a golden opportunity to equalise when Kane headed over from four yards out.
Newport goalkeeper Joe Day was not as busy as he might have expected to be, but made a crucial save when finally called upon on 67 minutes as he blocked a shot from substitute Son Heung-min with his legs.
Tottenham were far from their creative best in open play but managed to snatch an equaliser from a set piece as Son flicked on at the near post for Kane to apply a simple tap-in for his 30th goal of the season.
Newport survived the final eight minutes plus stoppage time to earn a replay which Spurs will be relieved to have salvaged from a difficult, and almost humiliating, trip to Rodney Parade.
NEWPORT (4-4-2): Day; Pipe, Demetriou, White, Butler; Willmott, Bennett, Tozer, Ladabie (Dolan, 84'); Nouble, Almond (McCoulsky, 74')
TOTTENHAM (5-3-2): Vorm; Trippier, Foyth, Dier, Vertonghen, Walker-Peters (Son Heung-min, 45';); Wanyama, Sissoko, Dembele (Davies, 81') ; Llorente (Alli, 66'), Kane