Thomas Mueller pounced on a deflected cross to put the visitors ahead in the 43rd minute and Dutchman Arjen Robben doubled the lead early in the second half as the 2013 champions took control of the tie.
The Italians looked dead and buried after an hour with the Germans having close to 70 percent possession and Juve desperately waiting for a rare chance to break.
Yet last season's finalists, who have been in superb form in recent months, staged a scintillating comeback.
Paulo Dybala scored in the 63rd minute and substitute Stefano Sturaro levelled with 14 minutes remaining to leave all to play for in the second leg after a frustrating finish for the Germans.
"We can live with this result and return home having a good starting position for the second leg," Bayern captain Philipp Lahm told reporters.
"Juventus are a good team but it was our own mistakes that got them back in the game. We played great for an hour."
Bayern shot out of the blocks in search of a quick goal and were almost rewarded when former Juve player Arturo Vidal's powerful volley forced 38-year-old keeper Gianluigi Buffon to palm his effort round the post.
Mueller then missed an excellent chance when he failed to tap in from two metres after being teed up by Robert Lewandowski's superb assist.
The Germany international, however, proved why he is such a prolific goalscorer, when he was in the right place to slot in from 10 metres for his sixth goal in seven Champions League matches this season, after Juve failed to clear a ball in the box.
Robben doubled their lead 10 minutes after the restart with his first goal of the year, a trademark move by the Dutch winger, who cut inside from the right and curled a left-footed effort into the net.
Dybala, however, made the most of a bad clearance by Bayern to beat Manuel Neuer on the hour and inject new life into the game and trigger a frantic Juve search for an equaliser.
Juan Cuadrado almost got it minutes later but Neuer calmly stood his ground to block the Colombian's shot. The keeper, however, could do nothing when Sturaro beat his marker to slide in and level.
"We knew it would have been tough, but I think we did well tonight," Dybala said.
"We gave everything we had, and we never gave up. It is going to be very difficult in Munich, but we know that we can make it with our heart, especially if we play like we did in the second half tonight."
(Reporting by Karolos Grohmann; editing by Toby Davis)
Share
