Aritz Aduriz's 32nd goal of the season had put Bilbao in control on home soil, but the holders of the competition will enter next week's second leg with an advantage after Iborra added to Timothee Kolodziejczak's 56th-minute leveller.
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Bilbao were without experienced midfielder Raul Garcia through suspension, while key centre-back Aymeric Laporte once again missed out after suffering a serious leg injury during the recent international break.
As for Sevilla, Kevin Gameiro was picked to lead the line ahead of former Bilbao striker Fernando Llorente, while Coke and Steven N'Zonzi returned at right-back and central midfield respectively for the holders.
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After a bright start from Bilbao, the first half-chance of the match came to Sevilla in the 11th minute, but Gameiro could only fire wide of Iago Herrerin's goal after finding space at the far post.
A slip from early substitute Federico Fazio allowed Williams time and space inside the Sevilla box, but the attacker's low cross was just too weak for Aduriz and Adil Rami was across to make a key clearance in the 15th minute.
Ever Banega then worked a yard of space outside the Bilbao box in the 22nd minute before releasing a powerful strike, but Herrerin was on hand to make a smart stop, before claiming the resulting corner.
Comfortably the clearest chance in the opening 30 minutes came to Williams, who had been sent through by Oscar de Marcos, but Soria was out to make a smart save at the feet of the 21-year-old.
Less than two minutes later, Sevilla had their best opportunity of the first period, but Vitolo, who had been put through by Michael Krohn-Dehli's pass, could not beat Herrerin from close range, before De Marcos headed a Williams cross wide of the post down the other end.
Bilbao had a golden chance to take the lead five minutes before the break when Benat delivered a free kick onto the head of Aduriz, but the Spanish international's effort hit the base of the post. De Marcos also came close for the home side late in the half, only to see his close-range strike deflect just wide for a corner.
Bilbao made the breakthrough in the 48th minute, however, when Aduriz headed a Muniain cross past Soria from close range to move onto nine European goals for the season and 32 in all competitions.
The home side came close to a second just five minutes later when Benat delivered a free kick onto the head of Xabier Etxeita, but the centre-back's effort brought a smart stop from Soria.
Sevilla were level in the 56th minute though when Banega took advantage of a poor header from Muniain before setting up Kolodziejczak, who turned inside the Bilbao box before placing his strike into the back of the net.
Both managers made changes as the final 20 minutes approached, with Yevhen Konoplyanka introduced for Sevilla and Inigo Lekue, Markel Susaeta and Borja Viguera all entering the frame for the home side.
Konoplyanka cut inside De Marcos ten minutes from time before attempting to pick out the bottom corner, but the Ukraine international missed the post as it remained 1-1 entering the final stages at San Mames.
Sevilla found a vital second goal in the 83rd minute, however, when Iborra turned a low Gameiro pass into the back of the net on the run. The midfielder looked slightly offside, but the flag stayed down as Sevilla celebrated their first away win of the season in any competition.