The draw in Group F, their second in a row, gives the world's 78th-ranked team a chance to reach the second round and stun the soccer world if they beat Paraguay on Thursday.
"We're still in with a chance in the group and that will surprise 99.9 percent of people," Herbert told a news conference.
"It's the most incredible result we have had. It's the biggest thing we have achieved in the game. We go into round (game) three with as good a chance as anyone."
Shane Smeltz gave New Zealand a shock seventh minute lead before Vincenzo Iaquinta equalised from the penalty spot after 29 minutes and a drab Italy were kept at bay fairly comfortably for the rest of the game at the Mbombela stadium.
"No one would remotely have given us a chance tonight but we did it. People have been saying we have amateurs in the side and shouldn't be at this World Cup."
New Zealand do not get a lot of practice against top sides having qualified from Oceania after facing the likes of New Caledonia and then beating Bahrain in a play-off.
"We were playing the four-times world champions, some people thought we shouldn't be part of this World Cup with the way we qualified," added Herbert, who played at the 1982 finals in their only previous appearance in the tournament.
"There will be a lot of excitement back home. We were always daring to dream at this World Cup and anything is possible."
‘I don’t think Liverpool would look at Ollie Watkins, a striker isn’t a pressing issue for them – it’s Arsenal who need one’ Former Reds star explains why his old club don’t need an out-and-out forward this summer
‘He’ll still be playing at 40 at a good level because he’s in such good shape and looks after himself so well. He does everything to be a top professional’: Ex-Liverpool coach insists Mohamed Salah has plenty more miles in the tank
