The hosts enjoyed the best chances of the first half, almost taking the lead in the 20th minute when Aritz Aduriz danced into the box only to curl his shot wide of the target.
It was Madrid who struck first in the match, taking the lead 20 minutes after the restart when Ronaldo picked out Jese Rodriguez inside the box, allowing the youngster to finish from close range.
However, an inspired substitution changed the game less than 10 minutes later as Ibai Gomez fired a fine strike into the bottom corner having been on the pitch for less than 60 seconds.
Things then got even worse for Madrid as their talisman Ronaldo was shown a straight red card after the referee adjudged that he had raised a hand to the face of Carlos Gurpegi.
That was the catalyst for Bilbao to push on for a winner, and they almost got it with just three minutes left when Ibai was denied by a fine stop from Diego Lopez.
However, Madrid held on to claim a point that keeps them third in La Liga but draws them level on points with second-placed Barcelona.