Aritz Aduriz set the 2008 cup winners on their way to a 3-0 quarter-final, second leg victory from Piatti's pull-back after 25 minutes before the provider turned scorer five minutes later, sprinting clear to net in a one-on-one.
The Argentine midfielder wrapped up the win at Levante's Ciutat de Valencia stadium near the end after the hosts' new signing Oscar Serrano was sent off on his debut, 18 minutes from time.
Valencia, who are third in La Liga, take on European and Spanish champions Barca next Wednesday in the semi-final, first leg tie at the Mestalla.
Last year's runners up Barcelona held on for a 2-2 draw at home to beat their great rivals and holders Real Madrid 4-3 on aggregate on Wednesday.
The second semi sees 23-times cup winners Athletic Bilbao take on giant-killers Mirandes of Spain's Segunda B, who have already put out top-flight Villarreal, Racing Santander and Espanyol.
It is only the second time a team from the third tier of Spanish football has reached the cup semi-finals and they have home advantage for the first leg next Tuesday.
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