The hosts went close to getting the breakthrough after 14 minutes through Inaki Williams, who dragged his close-range effort wide of Thibuat Courtois' goal.
Bilbao looked impressive on the counter and they nearly went ahead on a break, although Berchiche could only fire straight Courtois on 25 minutes.
The home side were not to be denied in the 32nd minute as Williams sent a cross into the box, where Iker Muniain was on hand to bundle home.
Luka Modric looked to restore parity for the European champions before the break, but the Croatian's long range effort was palmed away by Simone.
Second-half substitute Isco brought Los Blancos after 63 minutes, as he slotted home from close range after being teed up by Gareth Bale.
Sergio Ramos had a chance to put the visitors ahead in the 74th minute, but the defender blasted his free-kick over the crossbar from a very threatening position.
Sadly, the European champions could not carve out another opening and had to settle for share of the spoils on the road.