James Rodriguez scored a penalty 10 minutes from time to set up tense finale in Fortaleza, but Luiz Felipe Scolari's side held out for the victory to book their place in the last four of the tournament.
Oscar told reporters: "We played well - we deserved the 1-0 lead in the first half, or even more. We started well in the second half, wasted some chances, scored for 2-0 and then due to a mistake gave away a penalty.
"We knew how to control and win the game, our passing was much better today and we created many chances, our whole team played well and when everyone plays well, it's much easier. Our team is very good and we will do everything to win this World Cup."
Brazil will face Germany in their semi-final clash next week.