Igor Gabilondo gave the hosts the lead just before half-time as he stretched to fire home Andoni Iraola's cross into the middle.
But Villarreal were back on level terms eight minutes after the break as Giuseppe Rossi drove into the box and laid the ball across for Nilmar to finish at the back post.
The hosts were then reduced to 10 men with 10 minutes to go as Borja Ekiza was shown his second yellow card for bringing down Nilmar as Villarreal counter-attacked, but Athletic held on for a point.
Athletic had been poor in the midweek defeat at Malaga and Bielsa responded by making two changes with Fernando Amorebieta returning from suspension and Jon Aurtenetxe coming in for Ander Iturraspe and Oscar De Marcos.
However it was the visitors who started the brighter as Rossi teed up Borja Valero to fire over the bar before the Italian was brought down on the edge of the area but again Borja wasted the opportunity created by his team-mate, striking the resulting free-kick into the wall.
As in many of their games this season Athletic were enjoying plenty of possession without creating a great numbers of opportunities.
But they did go in front two minutes before half-time as Iraola burst down the right and with the Villarreal defence attracted to the run of Fernando Llorente, Gabilondo was free to shoot past Diego Lopez.
Rossi and Nilmar had both scored in the midweek win over Mallorca and their partnership was looking back to its potent best as the pair combined again for the equaliser.
Rossi was the creator as he strode into the area and cut the ball across for the Brazilian to score his second goal of the season.
Athletic had chances to go back in front, most notably from two Javi Martinez headers that were well saved by Lopez, but they were reduced to 10 men when Ekiza stopped Nilmar in his tracks as he bore down on the penalty area.
Villarreal enjoyed the bulk of possession from then on, but with Rossi replaced by Marco Ruben they did not create a chance worthy of winning the game and both sides settled for the point.