Through the brilliance of the Portuguese star, the capital outfit did what the hosts could not, taking their chances when presented to edge out a hard-earned victory.
But the scoreline did little to tell the story of the game, particularly the first half, as los Merengues were put under considerable pressure at times by the intense attacking style that Marcelo Bielsa's team are famed for.
Cristiano Ronaldo reached a half-century of goals for the season and needed only two minutes to break the deadlock in stunning fashion, before heading home another on 68 minutes to provide some much-needed breathing room for his team.
And Gonzalo Higuain arrowed home a third goal shortly after his arrival as a substitute to put any lingering hopes of a Bilbao comeback to rest, and restore Madrid's three-point cushion over third-placed Atletico Madrid, who were victorious over Granada earlier in the day.
What the Basques really could have done without was a poor start, but they made just that, when Ronaldo belted a 25-yard free kick into the top corner of Gorka Iraizoz’s net.
But it was hardly a sign of things to come. Bilbao would soon work themselves onto the front foot through their energetic attacking play that caused the Madrid defence no end of trouble.
Markel Susaeta was growing into the game with every passing moment, and with 19 minutes gone, drifted into a great shooting position on the right, but was denied by Diego Lopez at full stretch.
The home side continued to keep Madrid penned back in the first half. Their attacking players barely saw the ball and were in fact forced into some last-ditch defending - Angel Di Maria teaming up with Xabi Alonso to rob Susaeta from close range.
The Argentine though would almost make headway at the other end just before the interval, driving into the area on the counterattack and ignoring the option of a Karim Benzema cutback to smash towards goal, but he was thwarted by the crossbar.
But Ander Herrera would nearly have the last laugh at the very end of the half when he lashed an effort towards the top corner from the edge of the box that shifted inches wide, with Lopez appearing beaten.
Madrid showed signs of improvement early in the second half when Marcelo’s cross was beaten away by Iraizoz, who also recovered in time to stop Benzema’s follow up.
But the visitors could not quite build on the momentum earned from that opening, and walked a tightrope as the second-half progressed, with Bilbao still harbouring the potential to carve them apart.
However, with 22 minutes to go, relief would arrive for Madrid from the head of a familiar source. Alonso’s free-kick was met emphatically at the far post by Ronaldo, who soared above Jonas Ramalho to steer home.And eight minutes later, Madrid would put the icing on the cake. Higuain, who had been on the pitch for barely five minutes, angled home a fine effort into the corner after Ronaldo's flick freed him for the shot.
Substitute Jose Callejon could have added a fourth goal late on after wriggling into a great position inside the area, but steered his finish within a whisker of the post.