Atletico Madrid became outright leaders for the first time since their double winning season in 1996 earlier in the day with a 4-0 win over Real Sociedad on Sunday.
The draw moves Real level with Barcelona, who suffered a shock defeat at home to Valencia on Saturday, three points behind Atletico.
It could have been so much better for Madrid as after a difficult opening 45 minutes, Jese Rodriguez looked to have them on the road to a 11th consecutive victory when he prodded home Ronaldo's cross 25 minutes from time.
However, after Ibai Gomez had spectacularly equalised, Ronaldo was shown a soft red card for lashing out at Carlos Gurpegi.
"I think it is difficult for me to judge because I wasn't there but the images show that it was an exaggerated reaction," Real boss Carlo Ancelotti said about Ronaldo's red card.
"It could have been different because we scored first and then controlled the game well.
"Bilbao scored a fantastic goal, but, in general, I personally am satisfied with how Real Madrid played."
Ancelotti's men could have been in front after just 20 seconds as Ronaldo took advantage of a poor clearance by Gorka Iraizoz, but his effort flew inches wide.
Thereafter, it was Athletic who had the upper hand in the first-half and should have been in front when Aduriz shot wide when clean through on Diego Lopez.
Iker Muniain also passed up a great chance to open the scoring when he miscontrolled when unmarked inside the area.
However, Madrid responded after the break and Iraizoz was forced into two flying saves to deny Luka Modric and Karim Benzema.
The visitors did finally find a way through 25 minutes from time as Benzema fed Ronaldo on the left and his low cross was turned home on the stretch by Jese.
Real's lead lasted just nine minutes as Gomez made a stunning introduction as a substitute as with his second touch he fired an effort from 20 yards in off the inside of the post.
Madrid were then left to hold on for a point when Ronaldo saw red after a coming together with Gurpegi after having a penalty appeal turned down.
Gomez was denied a second by a brilliant save from Lopez, whilst the Madrid keeper also had to be alive to hold Aduriz's header as the spoils were shared.
Earlier, Atletico moved top on an emotional evening at the Vicente Calderon, as former players and fans paid their tributes to former Spain and Atletico player and coach Luis Aragones, who died on Saturday, before the match.
The Atletico fans remained silent for the first eight minutes before beginning to chant Aragones' name in a tribute to the number he wore as a player with Los Rojiblancos.
And Atletico boss Diego Simeone was pleased with how his side dealt with the emotion of the evening.
"When someone like Luis leaves us there is pain and sadness, but today the fans, the game and that silence for the first eight minutes before they sang his name moved me. From above he would be wearing red and white and happy with the bravery we showed.
"Real Sociedad made it difficult without creating great chances. Defending didn't worry me, but the problem was when you don't attack and it was difficult for us to get out in the second-half.
"The changes gave us some freshness. We had the ball more and we managed to get going again. It is a win against a very difficult opponent."
After a quiet start, David Villa, the competition's top scorer when Aragones led Spain to the European Championships in 2008, fittingly opened the scoring as he swept home Diego Costa's low cross.
Villa's joy didn't last long, though, as he had to be substituted just two minutes later after pulling up with a muscular injury.
Sociedad enjoyed the majority of possession after the break as Atletico decided to sit on their lead, but the Basque side failed to seriously test Courtois.
Costa made sure of the three points 18 minutes from time when he coolly slotted Raul Garcia's through ball into the far corner.
Atletico struck again just two minutes later as Miranda rose highest to power home Koke's corner.
And Diego Ribas rounded off the scoring on his return to the club after completing a deadline day loan move from Wolfsburg when blasted high into the roof of the net three minutes from time.