Despite going behind, Bilbao dominated the rest of the half as Markel Susaeta nearly found the net with a 20-yard lob. However, his effort drifted over goalkeeper David de Gea and just wide of the post.
Andoni Iraola also came close for the visitors soon after when his near-post drive was kept out by De Gea.
Bilbao scored a deserved equaliser a minute before the break as Fernando Llorente powered a header in from Susaeta's cross.
The Spanish side went on the attack instantly after the interval, with De Gea having to push away ferocious shots from Iker Muniain and Llorente.
The away side were rewarded for their attacking intent in the 72nd minute as Oscar de Marcos fired in a volley following a crisp passing move around the penalty area.
Bilbao's third came in controversial circumstances as the referee seemingly gave the visitors a free kick when Patrice Evra's boot came off. United, who were on the attack when the whistle blew, were expecting their opponents to give them back possession only for Bilbao to capitalise on some poor defending and score through Muniain.
The hosts managed to grab a second goal in injury time as Rooney converted from the penalty spot after De Marcos had handled.