Turkey midfielder Turan slotted in the first goal at the Nou Camp from Messi’s through pass in the 10th minute.
Barcelona goalkeeper Claudio Bravo then saved Vicente Iborra’s penalty after Samuel Umtiti handled.
Turan chipped in his second shortly after the break, with Messi guiding in a sublime header for the third.
It is the first time Barcelona have won the Spanish Super Cup since 2013.
The two-legged match usually pitches the league champions against the Copa del Rey winners. However, as Luis Enrique’s team won both trophies last season, Sevilla took part as Copa del Rey runners-up.
Both sides begin their La Liga season on Saturday, with champions Barcelona at home to Real Betis, while Sevilla, who finished seventh last May, host Espanyol.