Athletic will face Atletico Madrid in an all-Spanish Europa League final in Bucharest on Wednesday night.
While the Basque side will go in as slight underdogs, Llorente has warned his side will be fuelled by a stellar European campaign that hit its highest point at Old Trafford in early March.
Bilbao stunned the English champions 3-2 in their own backyard in the first leg of their last-16 tie before completing the shock progression with a 2-1 home win.
They subsequently eliminated Schalke and Sporting Lisbon en route to Wednesday's showpiece, but Llorente believes the club owe it to themselves to make sure the memorable Old Trafford success was not in vain.
"We have really had a great season in Europe, and that makes us even more determined to win the final," the Spain striker told reporters.
"By winning it, we want to make sure the incredible win at Old Trafford will not only be remembered as a great night during a positive European campaign which we didn't win."
Bilbao could yet secure a rare double, with a meeting against Barcelona in the Copa del Rey on May 24 also to come, in a run that has been credited in large part to boss Marcelo Bielsa.
The Argentinian has galvanised a team including the impressive attacking threat of Llorente and Iker Muniain in the cup competitions, if not the Primera Division where they sit 10th.
"One year ago, nobody would even have imagined that we could enjoy such a successful campaign. We now have to rise to the occasion, and end the season on a high," Llorente said.
The 27-year-old knows that his side face a tough task against Atletico who have themselves been rejuvenated since the arrival of Diego Simeone, in place of Gregorio Manzano, in December.
The capital club are unbeaten in their past seven games - compared to Bilbao who have won just four times in 11 - to lift them up to verge of a Champions League place.
"We know Atletico really well. They have changed a lot since the arrival of Simeone," Llorente said.
"I don't know how he has done it, but he has done a great job. They have been very impressive in Europe."
Wednesday's game will be Atletico's second Europa League final in three seasons after they beat Fulham 2-1 two years ago and they will again be expected to triumph given their big-game experience.
"When it comes to matches like this, those who've already played in one know what to expect, but that guarantees you nothing," Simone told www.uefa.com.
Simeone is also wary of Athletic's attacking threat, despite boasting the competition's equal-leading scorer Radamel Falcao, who has netted 10 times.
"They are a team with a lot of intensity, a direct style of football, quick and brave - a team that isn't afraid to play their way, even in away matches," he said.
"They've played like that throughout the whole season. And knowing the team and their coach, I know they will be the same in any game, whether it's a friendly or a final."
Atletico will have defender Luis Perea available after recovering from a thigh problem but Tiago is suspended while full-back Silvio is out for the season.
Bielsa rested a number of his key names at the weekend, including Llorente and Muniain, and should field a near full-strength line-up with only defender Fernando Amorebieta (calf) facing a late fitness test.