Arda Turan came off the bench to score the winner in Saturday's La Liga clash after Tiago headed the visitors in front and Real had equalised through a Cristiano Ronaldo penalty.
The result represented Real's second defeat in succession and prompted some fans to boo the players at Santiago Bernabeu, with goalkeeper Iker Casillas coming in for particular negative attention.
But defender Pepe says the European champions will need all the support they can get as they attempt to kick-start their La Liga campaign.
"It didn't go for us," he told Canal+. "It was a tough game, we had a lot of the ball, but we didn't shoot enough from outside the box.
"We have to lift our heads up, we have to work. This league won’t be easy [to win].
"The supporters are always right, but we need our supporters behind us."
Real host Basel in the UEFA Champions League on Tuesday.
