Real, 3-0 up from the first leg in Turin, found themselves down to two Mario Mandzukic headers in the first-half and conceded a third when Blaise Matuidi netted after the break following a fumble by keeper Keylor Navas.
However, Medhi Benatia's challenge on Lucas Vazquez saw Oliver point to the spot at the death as Buffon protested furiously before he was shown red in the Spanish capital - Wojciech Szczesny brought on for a penalty that was blasted past him by Cristiano Ronaldo.
Veteran Juventus goalkeeper Gianluigi Buffon, in what was likely his last Champions League game, was then sent off for furiously complaining after the penalty decision.
The Serie A giants had fought their way back from a 3-0 deficit to pull level, but with extra-time looming Cristiano Ronaldo managed to clinch a insane win via the penalty spot.
'I don't understand why they are protesting, ' Ronaldo said after the game.
The penalty came as Vazquez was in position to score in front of the goal after Ronaldo headed down a cross into the area. Just after Mandzukic got his second goal, Raphael Varane rattled the crossbar with a header to which Buffon got his fingertips.
"It was a match in which we suffered", Ronaldo said.
Arguably the strongest reaction of all came from the Catalan media, though, who cover the great rivals of Real Madrid, Barcelona. "What happened at the end of the game won't change or take anything away from what he has done in the game", Zidane told his press conference after the tie in Madrid.
In the night's other quarterfinal, Bayern Munich advanced with a 0-0 home draw against Sevilla as the newly crowned Bundesliga champions defended a 2-1 advantage from the first leg.
Cristiano Ronaldo has scored 10 Champions League goals against Juventus - the most by one player against a one opponent in the competition.
Right on this note, let us look at the three things that went right for Real Madrid against Juventus.
"I am happy and we are in the semi-finals". We're in the semi-finals.
It drew parallels with Real boss Zinedine Zidane when his France playing career ended with a sending off at the 2006 World Cup final for a headbutt on Italy defender Marco Materazzi.
"Everyone can decide what they want, if it's a penalty or not".
"I'm disappointed that they say we're through because of the referee".
AS admitted the European champions went from "the panic to the semi-finals", while Marca simply believed: "It was a penalty".