The Basque side turned around a one-goal deficit from the first leg to win the semi-final 4-3 on aggregate and set up a Europa League final with Atletico Madrid in Bucharest.
Both sides had chances to win, with the game played at an astonishing pace throughout.
And it was appropriate that it was Athletic's talisman Fernando Llorente who prodded home with just minutes remaining to send San Mames into rapturous delight.
Markel Susaeta had given Athletic the lead early in the first-half after Llorente cleverly chested the ball into his team-mate's path inside the Sporting penalty area.
At that stage it seemed as though the electric atmosphere inside the stadium was set to overwhelm the visitors.
But Sporting recovered well with Anderson Polga forcing a smart save from Gorka Iraizoz, before Ricky van Wolfswinkel silenced the home crowd with a cleanly struck 43rd-minute equaliser.
The response from Marcelo Bielsa's side was swift and Ibai Gomez put Athletic back in front on the night and level on aggregate when he took advantage of Llorente's delicious drag-back pass.
The second half saw both teams hit the woodwork as Javi Martinez headed onto the bar for Athletic before Sporting's Emiliano Insua drilled a left-footed effort against the post from distance.
And just when it looked as though the game was heading for extra-time, Llorente prodded home to break Portuguese hearts.