All three Turkish teams, Galatasaray, Fenerbahce and Anadolu Efes, were also knocked out in the fiercely competitive second group stage and will not feature in the May 11-13 Final Four on home soil in Istanbul.
Real did their job in Siena but were undone by Gescrap Bilbao's 59-55 success at Unicaja Malaga in an all-Spanish clash, enabling Euroleague debutants Bilbao to join Siena in the quarter-finals.
Siena, Bilbao and Real finished tied on four wins and two defeats each in their group but the former two progressed thanks to their better head-to-head records with Real.
"What we achieved today is huge for our first season in the Euroleague," Bilbao coach Kai Sikaris Fotios said.
"I want to congratulate my players and staff but most of all I dedicate this success to our fans."
Bilbao's reward is a clash with favourites CSKA Moscow in the best-of-five quarter-final series, which also pitted holders Panathinaikos Athens against last year's runners-up Maccabi Tel Aviv while Siena will lock horns with Olympiakos Piraeus.
Double champions Barcelona, who last won the trophy in 2010, will take on Russians Unics Kazan who advanced to the last eight despite a 68-63 home defeat by Panathinaikos.
The winners of the quarter-final series, which starts on March 20, will progress to the Final Four.
Kazan's defeat presented Fenerbahce with a great chance to reach the knockout stage but they blew it with an 85-72 defeat at already-eliminated Emporio Armani Milan, whose American forward Malik Hairston shone with a game-high 27 points.
Fenerbahce, by contrast, shot only 28 percent from three-point range and also committed 14 turnovers as Milan bowed out with their pride intact.
"We are disappointed with our elimination but I'd like to congratulate Milan who played a very good game and had an outstanding performance from Hairston," Fenerbahce's Croatian coach Neven Spahija said.
"We came here under a lot of pressure to win this game and we couldn't handle it as we didn't play well at either end of the court."
Fenerbahce were joined on the sidelines by Istanbul arch-rivals Galatasaray, who lost their dogfight with Olympiakos in Athens for a berth in the last eight.
The Greek side prevailed 88-81 in a high-scoring affair thanks to a classy performance by playmaker Vassilis Spanoulis, who poured in 20 points including a treble of three-pointers.
"It's a great accomplishment for my team, a very young one which was rebuilt from scratch this year and coach (Dusan Ivkovic) did a great job," Spanoulis said.
The 68-year-old Ivkovic, also the coach of his native Serbia, was confident that his revamped outfit would improve quickly enough to win top honours.
"Time is on our side and we will be much better in the near future. I don't think we are the underdogs in our quarter-final series against Siena although they will have the home court advantage," said Ivkovic.
"Olympiakos always has big targets and we have to respect and protect this legacy."