Cristiano Ronaldo was once again the star of the show as Real Madrid survived a second-half fight back from Galatasaray to secure a dramatic 5-3 aggregate win that sees them remain in the hunt for a record 10th European crown.
Having secured a 3-0 win last week at the Bernebeu the tie appeared to be all over within the opening 10 minutes as Ronaldo latched onto a cross from Sami Khedira to poke the ball home after getting in front of Emmanuel Eboue.
Galatasaray appeared to be missing their top scorer Burak Yilmaz, who was serving a one-game suspension, as they struggled to create a clear opening in the first 45 minutes.
The Turkish side were granted a reprieve in the second half when Ronaldo somehow scuffed a shot wide, when it seemed easier to score, and within minutes Eboue drew them level on the night.
Wesley Sneijder then missed a glorious opportunity when he pulled his effort wide but he quickly made up for the miss when a superb first touch took him past the excellent Raphael Varane and he fired his shot low past Diego Lopez.
Didier Drogba then sublimely back-heeled a cross from Nordin Amraba into the far corner and Galatasaray now had 20 minutes to find two more goals.
Galatasaray attempted to lay siege on the Madrid goal but even with Alvaro Arbeloa receiving a second yellow card they were unable to find the net and Ronaldo fired his 11th goal of the competition in injury time to seal the victory.
Ronaldo now holds the record for the most goals scored in the competition by a Real Madrid player and he is desperate to now help the club win the Champions League for the first time since 2002.
"We want to win the Champions League - we know it's difficult but for me it's possible. We have to carry on and the most important thing now is to win the semi-final because we want to be in the final; after that anything can happen," Ronaldo told the official UEFA website.
"We have to keep our feet on the ground, though, as we'll be up against a tough team in the semi-final. Dortmund are a team I've always liked - one of the teams that I thought could go very far. There's us and them and we'll see who else comes through, but the most important thing is that we're there."
Over in Germany there was even more late drama as Borussia Dortmund produced two injury time goals to secure a 3-2 aggregate victory that sparked wild celebrations inside the Westfalenstadion.
Dortmund progressed to the semi-finals of the Champions League for the first time since 1998 but just minutes before the final whistle it had seemed almost impossible for Jurgen Klopp's side to continue in the tournament.
Twice they had fallen behind, first to an excellent finish from Spanish winger Joaquin, and then with just seven minutes remaining substitute Eliseu prodded in Julio Baptista's shot to give Malaga the advantage.
Robert Lewandowski had earlier drawn Dortmund level but no one could have predicted the drama that unfolded in injury time. Malaga goalkeeper Willy Caballero had been in inspired form all night but there was little he could do as the German side scored two scrappy goals.
Marco Reus gave his side the faintest hope when he capitalised on a defensive mix-up to level proceedings but few of the 65,000 fans could have predicted there would be another opportunity.
Another goal mouth scramble led to Felipe Santana poking the ball over the line to complete one of the most memorable comebacks in Champions League history.
"It's really hard to describe what went on tonight, but we're happy - we're into the semi-finals. Those final five minutes were absolutely unbelievable. We knew we had to turn the game and it was chaos at the end, but so what? We're in the semis and we're delighted," said Lewandowski.
"It was difficult after Málaga's second goal but we knew that whatever happened, we needed to do something. At 2-2 we were desperate to get the winner and when Felipe [Santana]'s goal went in it was amazing. As I say, it's difficult to describe what happened but it must have been unbelievable for the fans.
"We need to see what happens in the semi-finals now and see who we're up against, but that doesn't matter really. We're one of the best four teams in the Champions League."