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."
"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."
"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."
