Schalke beat defending champion Inter Milan 2-1 at home, advancing 7-3 on aggregate following its stunning victory last week in Italy.
Madrid never looked likely to give up its 4-0 advantage over Tottenham from the first leg, and a second-half howler from Spurs goalkeeper Huerelho Gomes allowed Cristiano Ronaldo to score the only goal in a 1-0 win for the Spanish club at White Hart Lane.
Madrid will face archrival Barcelona in the semifinals, while Schalke takes on Manchester United.
"Let's see what happens," Madrid coach Jose Mourinho said. "A semi is a semi and anything can happen." While Inter may have held out faint hope of overturning its deficit, Raul Gonzalez all but snuffed it out when he gave Schalke the lead in the 44th minute, leaving the Italians needing five goals to advance. While Thiago Motta equalized for Inter in the 49th, Benedikt Hoewedes completed the surprising rout by netting the winner in the 81st as Schalke reached its first Champions League semifinal.
At White Hart Lane, Madrid soaked up early pressure from Tottenham, which had a goal disallowed for offside and a penalty appeal waved off in the first half.
Ronaldo then deflated the hosts when he sent a swerving shot from nearly 30 yards (meters) that went straight at Gomes, who fumbled a routine catch and watched the ball go over his shoulder before bouncing slowly over the line.
The mistake was so glaring that even Madrid goalkeeper Ikar Casillas just sadly shook his head in sympathy rather than celebrate the goal.
It was Ronaldo's 40th goal of the season, and left Tottenham playing only to salvage some pride.
Striker Roman Pavlyuchenko had a chance to equalize in the 58th when Rafael van der Vaart picked out the Russian with a cross into the area, but his header went high. Jermain Defoe then tested Casillas with a hard shot from outside the area in the 63rd, forcing the Spain goalkeeper to dive low and deflect the ball out for a corner, but the result was never in doubt.
"We lost it to a better team but we're disappointed to go out," Tottenham winger Gareth Bale said.
In Germany, Schalke waited patiently for chances to counterattack, and Jose Manuel Jurado set Raul free with a ball through a huge opening in Inter's defense. The former Madrid striker calmly rounded Julio Cesar to take his Champions League record tally to 71 goals.
Motta slotted home the equalizer after Lucio found him at the far post with a header after a corner, but it was too little too late. Schalke used another counterattack to get the winner, with Raul sending a long ball behind Inter's defense for defender Howedes to drive past Julio Cesar.
