Former Arsenal man Emmanuel Adebayor took his tally against Tottenham to nine in nine matches with two headers either side of half-time after Crouch's early dismissal for two bookings.
Harry Redknapp's miserable night was completed by strikes from Cristiano Ronaldo and Angel Di Maria, leaving Spurs with a surely insurmountable task in the return leg at White Hart Lane.
Gareth Bale offered them brief glimpses of hope, but their numerical disadvantage, and a top-drawer performance from Real's attack, meant they never really looked like grabbing an away goal that would have given them a fighting chance of staying in the quarter-final tie.
Before kick off, the 3,000 plus Tottenham fans filed into the Bernabeu hoping that they would witness another epic night in what has been a remarkable debut season in the competition.
They were lifted by good news from the treatment room about Bale and William Gallas, who were both passed fit despite injury worries.
However, Tottenham were dealt a blow before a ball had been kicked when Aaron Lennon was replaced by Jermaine Jenas in the starting XI after falling ill in the warm-up.
Just four minutes and one second had elapsed before Real opened the scoring, Adebayor beating Jenas to head home a corner despite the best, not good enough, efforts of Heurelho Gomes and Luka Modric on the line.
A late challenge from Crouch on Sergio Ramos deep in Real's half earned him a booking soon after and the striker did not heed the warning as he saw red in the 15th minute following a late and needless challenge on Marcelo, again deep inside the hosts' half.
Rafael van der Vaart was pushed up front, but he looked isolated and frustrated, earning a booking for petulantly kicking the ball away as Real prepared to take a free-kick.
It looked like only a matter of time before Real doubled their advantage - Ronaldo firing a volley at Gomes, and Marcelo giving Vedran Corluka a torrid time at right-back.
However Spurs had a golden chance to draw level just before the half hour when Van der Vaart took a long throw from Bale on his chest in the box but was denied by a block from Ricardo Carvalho.
Another glimmer of hope came Spurs' way a minute later when Michael Dawson released Bale down the flank but he drove into the side-netting.
Spurs started to look more comfortable on the ball, but Real still looked far more dangerous and Adebayor was only marginally too short to head home after Ramos had nodded Marcelo's cross back across goal.
Soon after, Modric threaded Bale down the left once more and he looked to have beaten Pepe but the Portuguese player slid in, clipping the Welshman's feet and earning a booking.
Real were adamant they should have had a penalty five minutes before the break when Dawson charged down a volley with his arms raised but the referee waved play on and Spurs got to the interval still in the game.
Redknapp injected some pace into his attack by replacing Van der Vaart with Jermain Defoe, but Real's ascendancy was shown when Adebayor grabbed his second after angling a header past Gomes from Marcelo's cross.
Only a brilliant tip-over from Gomes prevented Adebayor from netting his third but the goalkeeper was made to pay for punching a cross after 72 minutes when the ball found its way to Di Maria, who unleashed a beautiful 20-yard curler past the Brazilian for the hosts' third.
Defoe saw yellow for a high foot on Ramos before Adebayor received a standing ovation when he was replaced by Gonzalo Higuain with 15 minutes left.
Real's strength in depth was shown minutes later when £56million forward Kaka came on for Di Maria towards the end while Corluka limped off to be replaced by Sebastien Bassong.
Mourinho barked orders to his players, urging to find a fourth and Ramos came close to obliging with a drive that whistled just wide.
The killer blow did come three minutes from the end when Ronaldo got the goal his performance deserved, rifling a 20-yard volley past Gomes, who again should have done better, from Kaka's cross.