Athletic, looking for their first major trophy since 1984, took a 54th-minute lead through Jon Aurtenetxe but Sporting drew level through Emiliano Insua before the impressive Diego Capel - a Spain international - drilled home the match-winner 10 minutes from time to down his countrymen.
The Portuguese outfit pushed hard for a third goal late on but Athletic managed to keep them out to leave the tie firmly in the balance ahead of next week's second leg in the Basque country.
Both sides looked keen to attack throughout the first half, but neither goalkeeper was seriously troubled as the period finished goalless.
Sporting settled the quicker and Emiliano Insua drew an early save out of Athletic custodian Gorka Iraizoz before Ricky van Wolfswinkel fired just wide.
Athletic's first decent opportunity arrived in the 12th minute, and really should have resulted in the opening goal of the game but the unmarked Fernando Llorente, usually so deadly in the air, powered his header from Markel Susaeta's corner a foot over the crossbar.
Susaeta and Ander Herrera saw long-range efforts go wide and saved by Rui Patricio, respectively, as Athletic sought a valuable away goal, but Sporting were looking dangerous on the break and twice came close following lightning-fast counter-attacks.
The first saw Joao Pereira drive in a rising shot from the edge of the box which appeared to strike the outside of the post, although Iraizoz looked to have it covered.
Then, moments after Van Wolfswinkel had seen a 20-yard effort go just wide, Sporting broke away again but Stijn Schaars, after being set up by Capel, fluffed his lines
Sporting also threatened first at the start of the second half but, after Insua had seen his cross-shot parried by Iraizoz, it was Athletic who took the lead.
Susaeta curled in a free-kick from the right and after Insua had failed to cut the ball out - slicing his attempted clearance behind him instead of away from danger - defender Aurtenetxe slotted in at the far post.
That was a huge blow for Sporting, but it could have got even worse five minutes later as Athletic hit the woodwork, Fernando Amorebieta latching onto a loose ball inside the area to drive in a shot that struck the inside of the post before staying out.
Susaeta then saw a looping header land on the roof of the net with Rui Patricio struggling as the visitors went in search of a potentially decisive second goal.
At that stage it looked like Athletic would be left holding all the aces for their return to San Mames, but with their final dreams slipping away, Sporting staged a superb fightback.
Van Wolfswinkel spurned a great chance to equalise in the 69th minute when he glanced a header wide from close range, but within six minutes of that miss the home side were on level terms.
Insua was the quickest to react to a bouncing ball in the Athletic area to guide a header just inside the far post.
Buoyed by that goal, Sporting started to lay siege to the Athletic defence and it was 2-1 four minutes later when Capel drilled in a low 25-yard shot that fizzed into the near corner.
Substitute Bruno Pereirinha, having replaced Capel, should have made it 3-1 in the 84th minute but curled just wide from the near the penalty spot as Athletic survived a late onslaught to limit the damage.