Ferran Torres scored the game's only goal in the 28th minute, sealing a 5-4 aggregate victory after the first leg ended in a 4-4 draw on February 25.
Polish striker Robert Lewandowski, who scored in the first leg, came on as a substitute for Torres in the 74th minute.
Barcelona will face archrivals Real Madrid in the final on April 26 in Seville.
Szczęsny, a former Arsenal and Juventus keeper, had a quiet night in goal, but made several key saves to preserve Barcelona's lead.
He joined the club in October as a replacement for Marc-André ter Stegen, who was ruled out for the season with a knee injury.
Lewandowski, meanwhile, joined Barcelona from Bayern Munich in 2022. The 36-year-old forward also serves as the captain of Poland's national team.
(gs)
Source: IAR, PAP, TVP Sport