Aritz Aduriz cancelled out a first-half Karim Benzema effort to leave the two teams level in the second period, but Casemiro's close-range effort in the 68th minute saw Los Blancos emerge victorious at San Mames.
The victory moved Real Madrid five points clear at the top of La Liga, and Marcelo is pleased that his side have entered the international break with a win behind them.
"A game here is always difficult, we're in the final stretch now. What the coach said to us, we carried it out and we secured the victory," Marcelo told reporters.
"We trained and prepared for the game and it worked out well. We knew what we had to do and we did it. Top of the table? It's important, we're only thinking about picking up points, not how many we are in the lead. It was important to keep going before the break."
Real Madrid will return to league action at home to Alaves after the international break.