Two goals from Fernando Llorente and one each from Oscar de Marcos and Iker Muniain earned Bilbao a Europa League quarter-final first-leg win at Schalke.
Raul earlier gave Schalke a 2-1 lead with his double, before the Spanish side fought back to earn victory.
Elsewhere, AZ Alkmaar shocked Valencia with a 2-1 victory, while Sporting Lisbon saw off Metalist 2-1.
And Atletico Madrid substitute Eduardo Salvio scored in the closing stages to earn a 2-1 victory over Hannover 96.
Bilbao's victory in Germany gives them a vital advantage heading into the second leg on 5 April. Llorente had given the visitors a 20th-minute lead before former Real Madrid striker Raul equalised two minutes later.
Raul had given Schalke the lead on the hour with a fantastic left-foot volley but Spain international Llorente showed why he is so highly rated across Europe with a headed equaliser in the 73rd minute.
Bilbao took control of the tie with eight minutes remaining after De Marcos converted from close range and Muniain sealed the victory with a fine finish into the roof of the net from the edge of the area.
Alkmaar caused the biggest shock of the night with their 2-1 victory over Valencia. Aston Villa-bound striker Brett Holman gave the Dutch side a first-half lead with a volley straight from a corner before Mehmet Topal equalised in the 51st minute.
And with 11 minutes remaining Maarten Martens made it 2-1 to the hosts, side-footing Holman's cutback from six yards out to earn Alkmaar victory.
Sporting Lisbon's win against Ukrainian side Metalist was more straightforward although the Portuguese side were made to work hard late on after conceding a 91st-minute penalty which Cleiton Xavier converted.
Earlier, Marat Izmailov's strike and Emiliano Insua's free-kick had given them a 2-0 lead.
Atletico Madrid needed a 89th-minute goal to seal victory over Hannover 96 with substitute Salvio curling a shot from the edge of the area into the top corner of the net to give them a first-leg advantage.
Before Salvio's winner, Falcao headed in his seventh Europa League goal of the season before Hannover 96 equalised through former Manchester United striker Mame Diouf.