Lewandowski struck just before half time when he finished from Gavi's pass.
Gavi got another assist in the second half as he set up Pedri, before Karim Benzema struck a late consolation.
The Frenchman's late goal was scant consolation for Real Madrid, who were a distant second best throughout.
Indeed, Thibaut Courtois prevented Barcelona from winning by an even bigger margin with a couple of smart saves.
The former Chelsea goalkeeper was tested early when he turned Lewandowski's sweeping effort onto the post and he also kept out Ousmane Dembele's powerful strike.
But after Pedri got his side's third midway through the second half with a tap-in there was never any doubt about the result as Barcelona won the Super Cup for the first time since 2018.
Real Madrid were just unable to handle the brilliance of Gavi who, alongside his two assists, became the youngest scorer in a Spanish Super Cup, with his goal surpassing the record of his current manager Xavi.
