Alaba's deflected 30-metre effort caught Juve goalkeeper Gianluigi Buffon off guard and drifted into the bottom corner to hand the visitors a nightmare start as they suffered a first-half mauling.
Thomas Muller added Bayern's second goal just after the hour in a meeting of the German and Italian league leaders after Buffon had trouble with a long-range effort from Luiz Gustavo.
"It was a good match but we are not through yet. We hardly allowed Juventus a chance and we had several good ones," Bayern's Philipp Lahm said.
Juventus playmaker Andrea Pirlo made several uncharacteristic mistakes as his overrun side lost their first European match for three years after an 18-match unbeaten run.
"We knew that we were playing a great team and we knew we were a decent side too but in the end they really deserved their victory," Buffon told Sky Italia.
"On the first goal one of our players, perhaps Vidal, closed the ball down, it touched him and I couldn't reach it. The ball took another trajectory and there wasn't a lot I could do."
Juve had won their last five games in the competition without conceding a goal but that record quickly fell.
Pirlo's attempted pass out of defence went straight to Bastian Schweinsteiger who found Alaba and the Austrian's speculative shot took a slight deflection off Arturo Vidal to leave Buffon out of position.
JUVE OVERAWED
Juventus appeared strangely overawed and Bayern, who smashed nine goals past Hamburg SV in the Bundesliga on Saturday, poured forward with their rivals continually losing possession in midfield.
Buffon made up for his earlier mistake by blocking a close-range shot from substitute Arjen Robben as the Dutchman looked set to score and Franck Ribery was also close in an attack which began when Pirlo's pass sold Andrea Barzagli short.
Robben, who had replaced the injured Toni Kroos after 16 minutes, squandered another opening when he side-footed wide after Muller pulled the ball back.
Juve suffered further let-offs when Ribery and Alaba fired low balls across the face of goal in successive moves that nobody got on the end of and Schweinsteiger shot over from the edge of the area after a throw-in.
The second half began with another Bayern foray when a Mandzukic shot was blocked by Buffon after a counter-attack and Alaba nearly scored again, this time with a low free-kick which Buffon managed to turn away.
Juve's resistance was broken again in the 63rd minute when Buffon parried Luiz Gustavo's free-kick to Bayern's Croatia striker Mandzukic who rolled the ball for Mueller to tap into an empty net and give the Germans the upper hand for the return.
