The La Liga heavyweights were all but sure of progress in the competition after a 4-0 win over the Spanish third-tier side in the first leg on January 3.
Los Colchoneros dominated the entirety of the first half at the Wanda Metropolitano thanks to superb goalkeeping from Lleida stopper Diego Rivas.
However, the hosts broke the deadlock just before the hour through Yannick Carrasco, who finished off a swift counter-attacking move to bury Costa's cutback into the visitors' net.
New signing Vitolo rounded off an ultimately comfortable win for the 10-time Copa winners with a well-taken volley at the near post which opened his account for the club.
ATLETICO MADRID (4-4-2): Moya; Juanfran, Savic, Hernandez (Montoro 46'), Gonzalez; Carrasco (Correa 61'), Fernandez, Gabi, Vitolo; Costa (Gameiro 61'), Torres
LLEIDA (4-4-2): Rivas; Nunez (Valiente 33'), Adrian, Satrustegui, Kandeh; Lopez (Martinez 57'), Gning, Molina, Huertas (Cheng 77'); Nierga, Felix