The Bianconeri had conceded the fewest goals in the competition prior to kick-off but their 490-minute European spell without conceding was brought to a sudden end when David Alaba struck after just 25 seconds.
Speaking to Sky Sport Italia after the game, Chiellini said: "We knew it was going to be a difficult match for us, Bayern played at an incredible tempo and we struggled to keep possession when we were trying to build our game from the back.
"We hoped to earn a better result tonight but there are 90 minutes left; we will have to play with more intensity, like Bayern did today."
Chiellini also admitted that he was concerned by the Bianconeri's lacklustre performance but said they will perform better in the return leg in Turin next Wednesday.
"We know we can play better than today and I'm sure we will. We will continue to dream and we will make sure that we won't have any regrets after the return leg."
Juventus captain Gianluigi Buffon added that Bayern deserved the victory and felt that his side were fortunate to escape with a two-goal deficit.
“It’s clear we had some certainties about ourselves, but there’s nothing to say tonight, as they thoroughly deserved the victory and had even more chances to hurt us.
“Our opponents were better on the night, that is the truth. At times we had less sharpness under their pressure and I tried to alternate passes with longer kicks."
The 35-year-old shotstopper also commented on Alaba's opening goal, claiming that a significant deflection off Vidal influenced the direction of the ball.
“Alaba’s shot got a deflection and I was moving towards the right, but the change of direction caught me out. I had already made a step to the right and the ball changed direction mid-air.”
Meanwhile Juve midfielder Andrea Pirlo, who did not enjoy one of his finer performances in a Bianconeri jersey - achieving a pass completion of just 51% accuracy, said he was not happy with his display.
"I am the first to say I’m not satisfied with my performance, as I made many mistakes in such an important match and it’s disappointing," he said.
"We now have to analyse why we had so many misplaced passes. Clearly we need something more in the second leg, as what we did tonight was not enough."
Juventus will look to overcome the two-goal deficit on Wednesday April 10 but they will have to do so without Arturo Vidal and Stephan Lichtsteiner who will miss out through suspension after picking up yellow cards.