Bilbao, with new coach Mane at the helm, went ahead from Luis Prieto's deflected free kick in the first half, but Real's Brazilians led the fightback after the break with the winner coming after the visitors had been reduced to 10 men.
Real moved into second place on 29 points, just one behind leaders and arch-rivals Barcelona who were held to a 1-1 draw away to Levante on Saturday.
Earlier Espanyol fought back from a goal down to beat Sevilla 2-1 at home, which meant the Andalucians slip back to third with 28 points.
Mali striker Frederic Kanoute had put Sevilla ahead from the penalty spot after 26 minutes, but former Spain striker Raul Tamudo engineered an outstanding equaliser in the second half.
Facing his own goal and with two defenders in attendance, Tamudo pulled the ball back and spun between his markers before beating the advancing Andres Palop with a cool near-post finish.
Striker Luis Garcia volleyed the winner with 10 minutes remaining and although Espanyol lost midfielder Albert Riera to a red card they hung on for the win.
Atletico Madrid are fourth on 24 points after beating Real Betis 1-0 away on Saturday, and are a point ahead of Real Zaragoza who fell 2-1 at home to Osasuna after a last-minute own goal from Leonardo Ponzio.
Further down the table, Celta Vigo came back from two down to draw 2-2 away at Real Mallorca while Paco Flores got off to a poor start in his first game at the helm of bottom club Gimnastic, who lost 2-0
Real Madrid's counter-attacking style has given them the best away record in the league but their problems at home were there for all to see in the Bernabeu as they laboured to break down Bilbao in the first period.
Brazilian forward Robinho sparkled intermittently, setting up Michel Salgado for a shot early on which Inaki Lafuente did well to save at his near post.
But it was the visitors who struck first in the 34th minute when Luis Prieto's fierce free kick deflected off the chest of Igor Gabilondo enough to put off Iker Casillas.
Real coach Fabio Capello brought on David Beckham and Ronaldo at half time, and the home side stepped up the pressure.
After 65 minutes Ronaldo broke the offside trap to control Sergio Ramos's long pass and fire home the equaliser.
Bilbao's task was made more difficult when striker Aritz Aduriz was sent off for leading with the arm into a challenge with Mahamadou Diarra in the 75th minute.
Seven minutes later Roberto Carlos scored with a low shot from outside the area, but Real needed Casillas to make sure of the three points when he turned away Andoni Iraola's goalbound shot with his leg at the end.