The French striker put in a magnificent display, netting a double and providing an assist for Mesut Ozil, who was also on target in a commanding performance from the champions
Karim Benzema scored twice, one either side of a Sergio Ramos header, to put los Blancos three goals up within 32 minutes, but a great strike from Ibai Gomez gave Jose Mourinho something to think about at halftime.
The hosts were unperturbed after the break, though, and they continued with their dominant attacking display in the second half. Indeed, Mesut Ozil quickly ended hopes of a Basque comeback before Sami Khedira completed the scoring in the 72nd minute.
The signs of a domineering Madrid performance were clear from the opening 10 minutes as Cristiano Ronaldo saw his header deflected narrowly wide before Gorka Iraizoz did well to hold Jose Callejon's powerful shot moments later.
But Marcelo Bielsa's side could not hold out for much longer and the deadlock was broken as soon as the 12th minute. Luka Modric's long pass forward was plucked down by Benzema and his delightful effort took a slight deflection off Jon Aurtenetxe which looped over the keeper from the top of box.
Bilbao was powerless to resist the rampant Merengues, who narrowly missed out on a second moments later as Gurpegi cleared Callejon's shot off the line.
Ramos saw a header turned over the bar by Iraizoz midway through the half but as the hosts maintained the attacking onslaught, he did not fail at the second time of asking.
Ozil's in-swinging free kick was met by Ramos who was left unmarked to bury his header in the bottom corner with 23 minutes on the clock.
It only took two minutes for Madrid to find another and it came courtesy of another superb strike from Benzema. With his back to goal, the Frenchman dropped a should to gain a yard on Mikel San Jose before bending a sumptuous effort on the spin into the far corner.
Mourinho's side almost netted a fourth when Callejon hit the post on the stroke of halftime but the Bernabeu was stunned as Bilbao countered and found the net. Markel Susaeta was afforded space on the right and his rolled ball into the area was drilled home by Gomez.
The Basque side was unable to gain momentum from the equaliser and 11 minutes after the restart Ozil rounded off a fine move when he fired Benzema's cross home from close range.
Madrid was giving a performance befitting of a champion and it remained in the mood for goals. Benzema was only denied his hattrick moments later as Aurtenetxe got back to clear his clipped shot off the line.
But in the 72nd minute the fifth goal finally came. Coming in from the wing, Alvaro Arbeloa laid the offloaded to Khedira on the corner of the box. The Germany international took a touch before rifling a low shot underneath Iraizoz and into the net.
Madrid was wholly worthy of the emphatic scoreline and it ensured that they remains in third position, eight points behind league leader Barcelona.
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