Juve needed just four minutes to take the lead in Turin as Arkadiusz Milik headed in, only for Joao Mario to restore parity with a first-half penalty.
David Neres secured a deserved three points for Roger Schmidt's side after the interval, with Dusan Vlahovic seeing a goal ruled out for offside and Bremer blasting over as Juve looked to snatch a late equaliser.
That left Juve with just one win in seven European outings against Benfica, who sit level on points with Paris Saint-Germain at the top of Group H after the Ligue 1 side overcame Maccabi Haifa 3-1 on Wednesday.
Defeat also marked the first time Juve have lost their opening two games in the competition's group stage, while it is the first occasion they have lost three games in a row in the European Cup or Champions League since September 1972.
Pressure continues to mount on coach Massimiliano Allegri after just one win in Juve's last seven games, and Bonucci offered an honest appraisal after yet another frustrating performance at the Allianz Stadium.
"The whistles are right. There is little to say. We have lost a game that we absolutely shouldn't have lost," the Italy international told Amazon Prime Video.
"It is right that we are booed. I am the captain who must put my face to it. We are going through a moment where we find it hard to do everything.
"I'm worried, there is nothing to hide. Unfortunately, we leave the game too often. I don't know why and that's the thing that worries me the most.
"We struggle to keep the games constant. Now there is little to say, we just have to shut up, work and look forward.
"I think it is a situation that needs to be changed certainly. We have many players out, we are always playing the same ones and you feel the fatigue. It is everyone's fault."
Juve remains without a point after two Group H games in the Champions League, and will look to make amends on the return to Serie A action at Monza.