Louis van Gaal's side were not backed to advance from the group stage at the start of the tournament as they faced off against Spain and Chile.
However, the Netherlands won the group at a canter comfortably winning all three of their matches, including a 5-1 hammering of Spain in their opening game.
Sneijder told FIFA.com: "When you win, the atmosphere is always fantastic. We were in a very tough group and before the tournament a lot of people were saying that we didn't stand much of a chance of progressing.
"But we always believed and [just before the World Cup] we started to play a new system.
"From that, the confidence began to grow more and more. If you're in a group with Spain, Chile and Australia and you come first, winning all three games and scoring ten goals, then you've done a fantastic job."
Van Gaal's side face Mexico in their second-round clash on Sunday.