
The Basque side will play Real Sociedad in last season’s postponed Copa final on April 4 before taking on Barcelona in this year’s a fortnight later.
Roger Marti opened the scoring for Levante after 17 minutes with a close-range finish.
Raul Garcia equalised from the penalty spot on the half hour after being wrestled to the ground by Oscar Duarte.
The Basque side looked the more likely in the second period, but was unable to find the breakthrough before Berenguer’s long-range effort deflected off Nikola Vukcevic and flew into the net on 112 minutes.
Bilbao coach Marcelino Garcia, who won the 2019 Copa while in charge of Valencia, could complete a unique hattrick of back-to-back cup triumphs.