The 40-year-old felt that his side paid their German opponents too much respect but still remains optimistic over qualification
The Spanish side were second best for large periods of the game and Pellegrino voiced his concerns with the way Los Che approached the game.
"It has not been a good game. We had too much fear at the beginning," the 40-year-old Argentine told reporters.
"Perhaps we showed them too much respect," he added. "That is one of the reasons for our defeat, but overall we were solid."
Pellegrino watched his side fail to create goalscoring opportunities in a toothless first-half performance, but was pleased with how they responded in the second period.
"We gave the ball away a lot, especially in the first-half. We had no support when we brought the ball out, and we were not in the right positions," he continued.
"In the second-half we improved, but against a team like Bayern mistakes like these make things very difficult."
"We pressed them better after the break and we know we can improve.
"We have some important players injured, but that is no excuse. We had chances to draw level when it was 1-0, but the second goal was crucial."
"The players have shown good character in a difficult game, against a great team. Anything can happen in the group stage and the other result tonight [Lille 1-3 Bate] leaves things very open."