Bilbao had fought back from 2-0 down to draw level before Higuain latched onto Fernando Gago's pass and finished superbly just before the hour mark for Real's winner.
"The good thing about this squad is that we all go at it 100 per cent and did not notice the absentees," said Higuain. "Ruud (Van Nistelrooy) has not been able to play, but we have others capable of scoring.
"The day has been positive because we played against strong opponents and won. The game has not been easy. I hope to continue my goalscoring streak the next round against Almeria, but the important thing is that the team continues winning."