Bayern took the lead after just 25 seconds in the Allianz Arena when Alaba's long range effort took a slight deflection off Arturo Vidal and flew past Gianluigi Buffon.
At the other end, Andrea Pirlo's long range free-kick went just over before Toni Kroos went off injured to be replaced on 16 minutes by Arjen Robben.
Robben forced Buffon into a low save soon afterwards and Bayern began to dominate, creating a whole host of chances.
Bastian Schweinsteiger's free-kick went just over before Robben shot narrowly wide after being found by Muller.
Muller then teed-up Schweinsteiger for an effort 15 yards out in the centre but his shot went narrowly wide of Buffon's left post as the hosts went in at the break just a goal up.
In the second period, Bayern continued to press with Mario Mandzukic shooting from the right but Buffon saving at his near post.
The Juve keeper then had to dive down well to his left to keep out a long-range free-kick from Alaba.
But Bayern got the second goal their play deserved via a Muller tap-in on 63 minutes after Buffon could only parry Luiz Gustavo's long-range shot.
Mandzukic reacted best and latched onto the loose ball to set up his team-mate to put Bayern in control.
Vidal had a shot saved before Buffon denied Muller to keep his side within two goals of their opponents ahead of the second leg in Italy next Wednesday.