Villarreal goalkeeper Sergio Asenjo saved Aritz Aduriz's 40th-minute penalty at the Madrigal before teenager Oliver Torres slipped the ball across goal for Tomas Pina to fire the home side ahead two minutes into the second half.
The match turned when Villarreal defender Gabriel earned a second yellow card and was sent off in the 66th minute and Aduriz atoned for his penalty miss with a fine headed equaliser seven minutes from time.
With 10 games left, Bilbao have 52 points in fourth, six ahead of Basque rivals Real Sociedad, who are fifth on 46 points thanks to Sunday's 1-0 win at home to Valencia.
Promoted Villarreal, who have impressed on their return to the top flight, have 45 points in sixth, one ahead of seventh-placed Sevilla.
"The penalty was a bit of a blow but we managed to regroup after their goal and get the draw," Aduriz said.
"We started to dominate towards the end and we are more or less satisfied.
"There is still a long way to go and we have to take things step by step."
Villarreal midfielder Torres, on loan from Atletico Madrid, said: "The penalty save gave us a bit of a boost and we came out strongly in the second half.
"We managed to get ahead on the scoreboard but the sending off changed things and you really notice when you have a man less against such a good team," added the 19-year-old.
"We got a point but we are disappointed because we wanted the win."
Real Madrid are three clear at the top on 70 points after they won 1-0 at Malaga on Saturday.
Atletico Madrid are second on 67 thanks to their 1-0 success at home to Espanyol and champions Barcelona, who play at Real on Sunday in the Clasico, are a point further back in third after they thrashed Osasuna 7-0 on Sunday.
- Sports & Recreation
- Aritz Aduriz
- Oliver Torres
- Real Sociedad