That gave Carlo Ancelotti's men a slim advantage heading into Germany next week, with both teams still very much in with a shot of meeting either Atletico Madrid or Chelsea, who drew 0-0 in the first leg at the Vicente Calderon.
The Bundesliga champions, who are also the title-holders, saw their bid to become the first team to retain the Champions League take a blow as they dominated possession but ultimately failed to muster a response at the Bernabeu. It was only their second loss in this season's tournament, and also represented a first defeat for Pep Guardiola at the venue in his coaching career.
The Germans looked surprisingly toothless and the majority of Goal Singapore readers have tipped Real to hold on in the second leg, with 71.7% indicating that their quality and Ancelotti's tactical mastery will see them progress to the final. The remaining 28.3%, however, believe that Bayern are powerful at home and can turn things around.
Real Madrid are themselves on a quest for La Decima, as they seek to win a record 10th European crown.