A resolute Shakhtar Donetsk defence held the German champions at bay throughout – Pep Guardiola's side managing just a solitary shot on target during the round of 16 first leg in Lviv.
Starting with Robert Lewandowski on the bench, Bayern looked to Arjen Robben for inspiration but the chances he did create were spurned by wasteful finishing.
Having thrashed Hamburg 8-0 in domestic action at the weekend, Bayern had been heavily fancied to win again, and Gotze is keen to get the Ukrainian side to the Allianz Arena to put things right.
"We had some problems in attack today," said Gotze. "We weren't able to play as we expected.
"But there is still a second game, which we aim to win, though the little details will be decisive. It would have been great if we had scored today."
Bayern frustrations were added to just after the hour as they saw experienced midfielder Xabi Alonso dismissed for a second bookable offence, spoiling the Spaniard's 100th Champions League appearance.
The two sides meet again on March 11 at the Allianz Arena.
"The result is quite dangerous," said sporting director Matthias Sammer. "A goalless draw is always a problem.
"Actually it sounds good, but it isn't easy.
"We were far from our best and we have to improve in the next weeks."
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