A fortuitous opener from Jon Aurtenetxe put Marcelo Bielsa's side ahead, but the hosts fought back and turned the match on its head with two quickfire strikes
After chasing the game for much of the first period, Jon Aurtenetxe put the Basque side ahead just after the break, but Emiliano Insua got his side level, before Diego Capel completed the turnaround.
It was the hosts who had the better of the early proceedings. Marat Izmailov had already sent a limp effort wide of the target when Insua forced a fantastic save from Gorka Iraizoz with a ferocious free kick.
The early assault of the Bilbao goal continued when Andre Martins pulled the ball across the face of goal, finding an unmarked Ricky van Wolfswinkel, who could only send his shot wide, when he should have struck the back of the net.
In the end, however, Marcelo Bielsa's men were almost gifted the opener. Markel Susaeta swung in a corner straight to the unmarked Fernando Llorente, who could only head over from a matter of yards.
After a frantic opening, the match trailed off as the first half wore on. Susaeta tried in vain to break the deadlock, dragging an effort wide when from just outside the area, before Ander Herrera tried to fool Rui Patricio with a looping effort which the Sporting 'keeper gathered easily.
Bilbao had now come into the encounter after a slow start, but they failed to make it count, with another tame Susaeta effort rolling kindly into Patricio's arms.
At the other end, the hosts were still threatening, particularly on the break. Joao Pereira struck the outside of the post from a break-neck counter-attack, before Van Wolfswinkel struck wide once more with a powerful effort.
With the last kick of the opening period, Ricardo Sa Pinto's men should have gone in ahead.
Capel launched a break from an Athletic corner, but when the ball came to Insua, who had rocketed forward from the back, the former Galatasaray man lost his footing, letting the opportunity slip away.
Sporting came out from the break in close pursuit of the opening goal. Pereira sent in a cross from the right which appeared to strike Fernando Amorebieta on the arm, only for the referee to wave away frantic Portuguese appeals.
The decision proved to be decisive. Seconds later, a Susaeta free kick fell took an unlucky deflection off Insua into Aurtenetxe's path, and the 20-year-old kept his cool to tuck the ball past Patricio and grab his side a vital away goal.
Soon after, another Susaeta set piece caused mayhem in the Sporting box. Anderson Polga sent a poor clearance into straight to Amorebieta, who drove a shot against the woodwork with Patricio stranded.
The Basque club had now taken control of the contest, and were able to pass the ball leisurely in the midfield, with the hosts visibly down-heartened by falling behind.
Another free kick was deflected straight to Iker Muniain, who miscued his volley and failed to threaten the 'keeper, before Patricio was forced to scramble across the face of goal to deal with a looping Susaeta header.
Despite the visitors' domination, Sporting should have been back on level terms with 20 minutes left to play.
Izmailov sent in a great cross which Van Wolfswinkel could only nod wide when he should have buried it.
The miss only served to fire the hosts on. Shortly afterwards, Capel crossed the right, and a tame clearance fell to Insua who rocketed into the box ahead of Susaeta and headed the ball into the far corner.
That was just the beginning. Seconds later, Capel collected the ball 20 yards from goal, and sent a pile-driver into the bottom corner, past the sprawling Iraizoz.
Van Wolfswinkel squandered another opportunity, lifting a left-footed effort over the bar, thanks in part to a brave challenge, before Andre Carrillo flashed an effort narrowly wide.
In the end, however, the Portuguese outfit were made to settle for a one-goal lead, which Bilboa, having netted an away goal, will fancy their chances of overturning in a week's time.