The Serie A giants had stormed back from a 3-0 first-leg deficit to level the scores at 3-3 on aggregate at the Bernabeu on Wednesday night.
English referee Oliver then awarded a contentious spot-kick in added time after ruling Mehdi Benatia had bundled Lucas Vazquez over in the box.
Juventus goalkeeper Gianluigi Buffon is shown a red print business card in nigeria by referee Michael Oliver (AP)
Juve players furiously protested the decision, with goalkeeper Buffon receiving a red card, and the 40-year-old pulled no punches in his assessment of the official.
Speaking to Sky Sport Italia, he said: “The referee lost the plot, I accept that.
“I won’t accept that a referee is in charge of a game like this with two such high-profile teams and doesn’t have the character, courage or ability to make calm decisions.
“He was clearly unprepared. Giving a penalty like that after a memorable match, sending off someone who has never been red-carded for dissent in his life in my final game, is simply not acceptable.
“If you do that then you are an animal, you have rubbish in place of a heart. We did something epic this evening and the performance was heroic.”
Benatia was also irritated by the decision but did praise the way in which Juventus fought back in Spain.
“It was an absurd incident,” the Morocco defender told the club’s website. “We’re really disappointed.
“It’s unbelievable that our captain isn’t allowed to speak to the referee – the red card was very harsh.
“It hurts to get knocked out like that but I’m proud of this team. We played a fantastic game tonight in terms of intensity and technique.”
Two Mario Mandzukic headers and Blaise Matuidi’s second-half goal had remarkably dragged Juve back into the contest before Cristiano Ronaldo’s late penalty sent Real through.
Juventus manager Massimiliano Allegri, though, was keen to draw a line under the incident with his side four points clear in Serie A with seven games remaining.
“There’s no use crying about it,” he said. “We’re disappointed but we need to get that out of our system because we’re back in league action on Sunday (against Sampdoria).”