The towering striker was the hero at a typically vibrant San Mames, setting up Athletic's first two goals and then pouncing to volley the crucial strike two minutes from time to give the Basques a 3-1 win and take them through 4-3 on aggregate.
"This is incredible," Llorente said, struggling to make himself heard over the fans celebrating wildly in the stands.
"It's the first time I have felt like crying on a football pitch," added the Spain international, whose nickname is the Lion King due to his mane of thick blond hair.
"That was without doubt the most important goal I have ever scored."
Victory for Athletic, chasing their first continental title, and the presence of their compatriots and 2010 winners Atletico Madrid, who comfortably held on to a first-leg lead to oust Liga rivals Valencia, in the May 9 showpiece in Bucharest may help ease Spanish pain in Europe this season.
The country's two heavyweight clubs, Real Madrid and Barcelona, were dumped out of the Champions League this week.