Valencia were only seconds away from reaching the final after taking a three-goal lead in the second leg against Sevilla at the Mestalla on Thursday, putting them 3-2 up on aggregate late on.
Sofiane Feghouli, an own goal from goalkeeper Beto and Jeremy Mathieu had put Juan Antonio Pizzi's men on the verge of completing a sensational comeback after losing the opening leg 2-0.
But a 94th-minute goal from Stephane M'Bia was enough to see Sevilla through on away goals after the semi-final tie finished 3-3 on aggregate.
"I have no words," Piatti said. "We feel very bad and we are very hurt.
"We tried, we gave everything and they scored against us in the last minute."
Valencia had already successfully overturned a first-leg deficit when they upstaged Swiss opponents Basel 5-0 to progress 5-3 on aggregate in the quarter-finals.
And the Spanish outfit almost replicated that feat in the semis but were cruelly bundled out by their La Liga counterparts.
"We are pleased with the work we have done, but it didn't amount to anything," he said.
"I want to thank everyone who came to support us."
Valencia's drought in European finals continues, with their last appearance in a UEFA club competition decider was back in 2004, when they upstaged Marseille to claim the UEFA Cup.
