© Getty Images
The visitors came close to opening the scoring a quarter of an hour into the tie when Arjen Robben unleashed a strike from the edge of the box, but it drifted just wide of the post.
Real were able to break the deadlock just a couple of minutes later when Fabio Coentrao sent in a low cross for Benzema, who was able to tap the ball home from close range.
Cristiano Ronaldo, who was returning from a four-game absence with a knee injury, almost added the home side's second when he tried to head in from an Angel di Maria delivery, but his finish went straight to Manuel Neuer.
Bayern dominated the possession, but Iker Casillas remained largely untroubled in the home side's goal for much of the second 45 minutes.
Casillas was called into action with just over five minutes still to play when Mario Gotze managed to round several defenders, but his finish was straight at the Real keeper.
Bayern continued to apply pressure inside the hosts' half in the closing minutes, but Real's midfielders dropped back to help out in defence, preventing them from getting a clear sight at goal.
Bale tried to create a second of the night for the Spanish side on the counter-attack late on, beating Dante with his pace before sending his shot into the side-netting.
Muller appealed for a penalty deep into stoppage time after taking a tumble under a challenge from Xabi Alonso, but Howard Webb did not believe that he was fouled, leaving the defending champions a goal down heading into next week's second leg at the Allianz Arena.