Cristiano Ronaldo scored a hat-trick to send Real Madrid through to the Champions League semi-finals with a 4-2 extra-time win and 6-3 aggregate success against 10-man Bayern Munich in controversial circumstances on Tuesday.

A bizarre Sergio Ramos own goal forced the extra period after Robert Lewandowski opened the scoring for Bayern from the penalty spot and Ronaldo equalised.

Marcelo twice cleared off the line for Madrid and Jerome Boateng did the same for Bayern in a thrilling game, which was tilted in Madrid’s favour by Arturo Vidal’s dismissal for two yellow cards, the second of which was harsh.

Ronaldo put Madrid level at 2-2 on the night in the first period of extra time when he scored his 100th Champions League goal, from a clearly offside position, before he wrapped up his treble with another disputed goal and Marco Asensio added a fourth.

Bayern coach Carlo Ancelotti was furious, criticising Hungarian referee Viktor Kassai after the game and asking for the introduction of video technology.

“I think we played the game very well, we deserved more. You have the decisions that penalised us a lot, Arturo’s card was not a card and then two goals for Cristiano were offside,” said Ancelotti.

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

Bayern began well, with Thiago Alcantara's close-range shot brilliantly blocked on the goal line by Marcelo. Arjen Robben lashed the rebound into the side-netting.

Madrid began to find their feet and were soon dominating, with the effervescent Marcelo key in both attack and defence.

Ramos nearly put the hosts ahead after the otherwise superb Manuel Neuer spilled a shot, but Boateng somehow blocked his effort on the line.

However, it was Bayern who took the lead in the second half when Casemiro fouled Robben in the area and Lewandowski dispatched the penalty.

Ronaldo levelled, heading beyond Neuer in the 76th minute from a fine cross by Casemiro, but Bayern restored overall parity within a minute when Ramos prodded into his own net after a scramble in the area.

With the game heading towards extra-time Vidal was sent off for a second booking after appearing to win the ball cleanly from Asensio, leaving Bayern with 10 like in the first leg when Javi Martinez was dismissed.

It gave Madrid the upper hand going into the added 30 minutes and they took advantage when Ronaldo scored the crucial second from Ramos's cross, despite being in an offside position.

Madrid added two more late on, Ronaldo appearing fractionally offside again for their third of the night, leaving Bayern exhausted and angry.

“The referee’s task is quite difficult, whether it’s for or against us I won’t get into it,” said Madrid coach Zinedine Zidane.

“More than the second yellow (for Vidal) or the offside goals, the six goals we scored over the two games means we deserved to go through in the tie.

“There is no luck in football, you must win, and we won both games.”