And Fallon told BBC Sport: "I personally believe that we can go through to the next round.
"We can kick on and do even better, that's the beautiful thing about our team - we believe we can do it."
New Zealand's draw with Slovakia has left all four teams in Group F on a point each after Italy's draw with Paraguay on Monday.
And with their ever first point at World Cup now safely in the bag, Fallon feels the Kiwis' next game against defending champions Italy is anything but a foregone conclusion.
He said: "I'll never go out onto a football pitch thinking I'm going to lose.
"I'll always go out onto a football pitch believing I can be the best and that we can beat the best, and I know that we've got a good enough team to do that.
"You saw with Italy [against Paraguay] that they struggled on crosses and diagonal balls and hopefully we can get into the game through that."
The 28-year-old striker also paid tribute to the fans at Plymouth Argyle, where he has played since 2007.
Plymouth were relegated from the Championship last season, but in Fallon the supporters have the first ever active Argyle player to appear at a World Cup finals.
He said: "I just want to thank all the Green Army for their support.
"It's been a tough two years but they've just got to keep believing as well.
"All of these things happen for a reason and with the Green Army going down I believe that they're in the rebuilding stage and it's like the phoenix from the ashes.
"It's just like New Zealand football, we've been in the ashes for a long time and we've been waiting to rise back up and today we've done it.
"But we've got to keep on believing and keep on working hard."
Source: BBC Sport
Source: BBC Sport