The League Two hosts did a real job on us in the first half and were eight minutes away from causing a major upset thanks to Padraig Amond's header seven minutes before the break.
But Kane popped up with a back-post tap-in with the wind blowing and the clock ticking down after Kieran Trippier's corner was flicked on by substitute Heung-Min Son to force a Wembley replay in this fourth round tie.
Son might have levelled earlier in the second period but for a vital save from Exiles goalkeeper Joe Day on a heavy Rodney Parade pitch.