The hosts went into the game unbeaten in their last six La Liga games on home turf and averaging over four goals a game, but they struggled to find a way through a stubborn Basque rearguard led by tormentor-in-chief Kepa, who demonstrated exactly why Real Madrid were keeping close tabs on him earlier in the season.
Bilbao were the first to threaten as Raul Garcia slipped in Mikel San Jose, but Keylor Navas made himself big at the near post to deny his effort from a narrow angle.
Cristiano Ronaldo almost opened the scoring when he met Dani Carvajal's cross from the right, but his header crashed back off the crossbar with Kepa a mere spectator.
With 14 minutes on the clock though the ball was in the net and it was the visitors who were celebrating. Clearly having not learned form their earlier scare, Real once again failed to close down Raul Garcia and he picked the perfect through ball for Iñaki Williams, who scampered clear and kept his cool to dink the ball over the onrushing Keylor Navas and into the back of the net.
At that point, the Kepa show began, as the Spanish international set about proving why he is be considered on of La Liga's top stoppers. A sharp low save from Ronaldo started him off, before he parried away Raphael Varane's header from a Toni Kroos free-kick.
Real pressed for long spell after going behind but were unable to level before the break as Kepa produced a sensational right-handed stop to deny Marco Asensio after he had wriggled away from three men down the right on the storke of half-time.
18 April 2018
- Continued Real Madrid pressure -
As the game moved into the second 45, the hosts continued to have the better of things, but barring a succession of corners they failed to really create any chances, with Iñigo Martinez a rock at the heart of the Bilbao back four.
With 64 minutes gone, the visitors squandered a great chance to make the game safe as Navas made a mess of a cross from the right, but Iñigo Cordoba's effort at the far post was blocked on the line by Carvajal, before Garcia crashed the rebound against the crossbar from 12 yards.
Following yet more Real Madrid pressure, the Basques were eventually made to regret that miss as Real found a leveller with just three minutes to go.
The Ball dropped to Luka Modric on the edge of the box in the wake of a corner and his low effort was flicked home by Ronaldo as it made its way through the crowd to earn the hosts a point.
18 April 2018
The draw leaves Real Madrid in third with 68 points from their 33 games so far, three points ahead of Valencia in fourth and just three points behind city rivals Atletico in second, though Diego Simeone's men do have a game in hand.
Bilbao, for their part, move into 13th, with 40 points.