The hosts will now face Colombia in the quarter-finals after winning through 3-2 on spot kicks following a 1-1 draw.
Scolari said: "We took upon ourselves this mission that we must be champions. If you make a promise you must deliver. This is what the players are doing.
"There are three more games to see if we can reach heaven."
Brazil have now lost just one of their last 16 World Cup games when scoring first.