Bayern created a string of chances in the first half and ended the game with 14 shots on target compared to the Italians' seven, but Louis van Gaal's side failed to break the Juventus defence.
"We did not win and it's not every week that we can say: 'We could have won' - simple as that," said van Gaal bluntly, while praising striker Miroslav Klose for a "very good performance."
"We have to score goals to win and we created so many chances to win, so it is a pity we didn't get a result and we are very disappointed. "But if we continue to play in such a way, it is only a question of time before the results go our way."
With Maccabi Haifa losing 1-0 at Bordeaux, Bayern stay top of the table only by virtue of a superior goal difference compared to the French champions, who they will meet in the next round of matches.
But the Champions League clash might have cost Bayern the services of Dutch winger Robben, who left the pitch limping after injuring his right knee just before the half-time whistle.
"I think it is a serious injury - it doesn't look good," said van Gaal afterward.
Van Gaal opted to start Germany star Miroslav Klose as the lone striker with Robben and Franck Ribery - dubbed 'Robbery' by the German media - on the wings in a 4-3-3 formation.
In defence, 20-year-old Holger Badstuber partnered Belgium's Daniel van Buyten at centre-back with Germany's Philipp Lahm and Holland's Edson Braafheid on the flanks.
Juventus coach Ciro Ferrara opted for his customary 4-4-2 with France's David Trezeguet partnering Italian World Cup winner Vincenzo Iaquinta up front. The sides had experienced mixed fortunes in their opening games, with Bayern winning 3-0 at Maccabi while Juve drew 1-1 at home to Bordeaux.
The Germans threatened in the opening minutes when 20-year-old midfielder Thomas Müller fired across the face of the goal, but his effort went wide, while Trezeguet picked up an early card.
Brazil's ex-Werder Bremen midfielder Diego then fired straight at Bayern goalkeeper Hans-Jörg Butt on 18 minutes, but seconds later Ribery beat two defenders only to float his shot over both goalkeeper Gianluigi Buffon and the crossbar.
Just before the half-hour mark, Müller broke through and went one-on-one with Buffon, but tried to play Robben into space and wasted the chance, with Klose heading over from the resulting corner.
Juve's Mauro Camoranesi then fired wide of the post while seconds later both Müller, Klose and Schweinsteiger all had chances against Buffon. Ribery was sent tumbling to the turf on several occasions by robust Juventus marking and referee Howard Webb finally lost his patience when he showed Camoranesi a yellow card on 36 minutes.
The game went flat in the second-half as Juventus playmaker Diego went off and van Gaal brought on Germany striker Mario Gomez to spruce up Munich's attack.
Iaquinta had arguably the best chance of the game to put the guests ahead when he mistimed his shot on 65 minutes and slammed his fist into the turf in frustration - it was one of those nights.
"We are not completely content, because we wanted to win here," said Diego. "Bayern played a good game, but we also had three or four good possibilities to win."