Bayern Munich left the Santiago Bernabeu furious with referee Viktor Kassai and his team of assistants after being knocked out of the Champions League by Real Madrid on Tuesday.

Real Madrid won 4-2 after extra-time to advance with a 6-3 aggregate success but Cristiano Ronaldo was offside for two of his three goals and a harsh dismissal for Arturo Vidal left a fuming Bayern with 10 men.

Chile midfielder Vidal was shown a second yellow card by Kassai in the 84th minute for what appeared a clean challenge on Marco Asensio, tilting the game in Madrid's favour as they entered extra-time trailing 2-1.

Real Madrid 4 Bayern 2 player ratings Show all 22 1 /22 Real Madrid 4 Bayern 2 player ratings Real Madrid 4 Bayern 2 player ratings Keylor Navas – 6 out of 10 Found wanting when called upon. Should have done better to prevent Ramos’ own goal. Real Madrid 4 Bayern 2 player ratings Dani Carvajal – 7 out of 10 His early effort from range may not have gone in but it set the tone for Madrid to have the better of the first half. Real Madrid 4 Bayern 2 player ratings Nacho – 7 out of 10 More solid than his defensive partner, and looks to be benefitting from gaining Zinedine Zidane's confidence. Real Madrid 4 Bayern 2 player ratings Sergio Ramos – 6 out of 10 Unfortunate to divert the ball into his own net, but it could have been avoided with a little more communication. Real Madrid 4 Bayern 2 player ratings Marcelo – 9 out of 10 Excellent. Superbly blocked Thiago’s effort early on, then produced a brilliant goal-line clearance after the break to deny Robben. Impressed going forward too, setting up Ronaldo in extra time. Real Madrid 4 Bayern 2 player ratings Luka Modric – 7 out of 10 Key to Madrid’s excellent spells of first half build-up. Ronaldo should have rewarded one particularly special pass with the goal it deserved. Real Madrid 4 Bayern 2 player ratings Casemiro – 7 out of 10 Careless in his challenge on Robben for the penalty. Otherwise, a solid presence disrupting Bayern’s forays forward. Real Madrid 4 Bayern 2 player ratings Toni Kroos – 5 out of 10 Struggled to match the influence of his fellow midfielder Modric. Real Madrid 4 Bayern 2 player ratings Isco – 6 out of 10 Failed to build on Saturday’s excellent showing at Sporting. Real Madrid 4 Bayern 2 player ratings Karim Benzema – 5 out of 10 Offered very little before being hauled off for the more impactful Asensio. Real Madrid 4 Bayern 2 player ratings Cristiano Ronaldo – 8 out of 10 A quiet start but then came another ‘clutch’ goal, just like his two in Bavaria. Two more followed thanks to poor officiating, but this was still a reminder that you should never write Ronaldo off. Real Madrid 4 Bayern 2 player ratings Manuel Neuer – 6 out of 10 Will be disappointed to have conceded six over the two legs, but was outdone by some poor officiating here. Real Madrid 4 Bayern 2 player ratings Philip Lahm – 6 out of 10 Struggled to cope with Marcelo, who got the better of him too often. We have come to expect more. Real Madrid 4 Bayern 2 player ratings Jerome Boateng – 7 out of 10 Heroic block on the line stopped Ramos from putting the hosts in front in the first half. Real Madrid 4 Bayern 2 player ratings Mats Hummels – 8 out of 10 Another defender who put his body on the line when it counted. Spectacularly blocked Kroos’ shot in the first half. Real Madrid 4 Bayern 2 player ratings David Alaba – 5 out of 10 His poor crossing let his teammates down when they were making headway early on. Real Madrid 4 Bayern 2 player ratings Xabi Alonso – 5 out of 10 Surprisingly poor in possession during his last Champions League appearance, with several wayward passes. Replaced by Thomas Muller deep into the second half. Real Madrid 4 Bayern 2 player ratings Arjen Robben – 8 out of 10 Made things tick in Bayern’s attack. Questionable role in the penalty, maybe, but his side's stand-out performer still. Real Madrid 4 Bayern 2 player ratings Arturo Vidal – 4 out of 10 His tackle was fair and should not have resulted in a red, but still, too often a liability. Should also have done better when slipped in by Robben soon after Bayern’s first goal. Real Madrid 4 Bayern 2 player ratings Thiago – 5 out of 10 Off the pace in the opening stages and failed to grow into the game. Real Madrid 4 Bayern 2 player ratings Franck Ribery – 5 out of 10 A fading force, unfortunately. Simply did not trouble Madrid’s backline with the regularity of Robben. Withdrawn for Diego Costa. Real Madrid 4 Bayern 2 player ratings Robert Lewandowski – 6 out of 10 Not his best night, despite cooly converting the penalty.

Ronaldo scored the next goal in the 105th minute from a clear offside position, and then sent Madrid 3-2 up on the night when fractionally offside.

“When they rob a game from you like that, it’s very tough. This error cannot happen in the Champions League,” said Vidal.

“When it was 2-1 they got scared and the referee started putting on his show. We played with one less man at home and here, too. The referee put us out of the Champions League.”

Ronaldo hit a hat-trick against Bayern Munich (AFP/Getty)

There were other contentious decisions on the night, including Casemiro avoiding a red card for a late foul on Bayern's Arjen Robben when already booked and an offside against Robert Lewandowski in the second half when the Polish striker was well placed.

“It is very harsh, two offside goals, the red card for Casemiro before mine,” said Vidal.

“The dressing room is very sad because we really wanted to go through but we have to keep improving and hope these things will not happen again.

“We are very angry, a game with that much intensity cannot be decided by the referee.”

Vidal was dismissed against Real (AFP/Getty)

Bayern coach Carlo Ancelotti agreed with Vidal and said in his media conference that the referees needed video help.

“Arturo’s card was not a card and then two goals for Cristiano were offside,” the Italian.

“We are not happy about this. In a quarter-final you have to have a referee with more quality, I think. It is time to introduce videos for the referees.”

Ancelotti wished Zidane all the best (AFP/Getty)

His Real Madrid counterpart Zinedine Zidane did not want to discuss the refereeing but he claimed Bayern's second goal could have also been ruled offside.

Lewandowski appeared to be fractionally offside in the build-up to Sergio Ramos’s own goal, causing panic in the area before the defender prodded the ball past Keylor Navas.

“They have told me their second goal was also offside. It goes both ways, that is football,” said Zidane.