The 27-year-old scored with a header after 72 seconds before Marco Asensio chipped in a brilliant finish for 2-0.
Bale rounded Sociedad keeper Geronimo Rulli to score again in the final minute for last season's runners-up.
Sociedad fans applauded throughout the ninth minute as a tribute to their former striker Dalian Atkinson, who died on Monday.
Atkinson wore the number nine shirt when he played for the club during the 1990-91 season.
Meanwhile, Atletico Madrid - who finished third in May - started their season with a 1-1 draw at home to promoted Alaves.
The hosts looked to have won it with an injury-time penalty from debutant Kevin Gameiro, after Fernando Torres had been pulled down in the area.
However, the visitors, who had previously shown little ambition, gained a point thanks to Manu Garcia's fine strike in the final seconds.
There were unpleasant scenes during Sporting Gijon's home game against Athletic Bilbao.
The referee had to halt the match briefly because of racist chants being aimed at visitors' forward Inaki Williams by some home fans.
Sporting won the match 2-1.
Match ends, Real Sociedad 0, Real Madrid 3.
Second Half ends, Real Sociedad 0, Real Madrid 3.
Goal!  Real Sociedad 0, Real Madrid 3. Gareth Bale (Real Madrid) left footed shot from the left side of the six yard box to the centre of the goal. Assisted by James Rodríguez.
Isco (Real Madrid) wins a free kick in the defensive half.
Foul by David Zurutuza (Real Sociedad).
Attempt saved. James Rodríguez (Real Madrid) left footed shot from the right side of the box is saved in the bottom right corner. Assisted by Lucas Vázquez.
Attempt missed. Mateo Kovacic (Real Madrid) right footed shot from outside the box is too high. Assisted by Casemiro.
Delay over. They are ready to continue.
Delay in match Gareth Bale (Real Madrid) because of an injury.
Yuri (Real Sociedad) is shown the yellow card.
Sergio Ramos (Real Madrid) is shown the yellow card.
Casemiro (Real Madrid) wins a free kick in the defensive half.
Foul by Willian José (Real Sociedad).
Attempt blocked. Xabi Prieto (Real Sociedad) right footed shot from the right side of the box is blocked. Assisted by Iñigo Martínez.
Delay over. They are ready to continue.
Delay in match Kiko Casilla (Real Madrid) because of an injury.
Corner,  Real Sociedad. Conceded by Marcelo.
Attempt blocked. Mikel Oyarzabal (Real Sociedad) right footed shot from the right side of the box is blocked. Assisted by Xabi Prieto.
Corner,  Real Madrid. Conceded by Iñigo Martínez.
Sergio Ramos (Real Madrid) wins a free kick in the defensive half.
Foul by Mikel Oyarzabal (Real Sociedad).
Substitution, Real Sociedad. Carlos Vela replaces Mikel González.
Attempt missed. Xabi Prieto (Real Sociedad) header from a difficult angle on the left is high and wide to the left. Assisted by David Zurutuza with a cross following a corner.
Corner,  Real Sociedad. Conceded by Daniel Carvajal.
Attempt blocked. Yuri (Real Sociedad) left footed shot from the left side of the box is blocked. Assisted by Markel Bergara.
Substitution, Real Madrid. James Rodríguez replaces Álvaro Morata.
Lucas Vázquez (Real Madrid) wins a free kick in the defensive half.
Foul by David Zurutuza (Real Sociedad).
Attempt missed. Willian José (Real Sociedad) header from the centre of the box misses to the left. Assisted by Zaldúa with a cross.
Corner,  Real Sociedad. Conceded by Sergio Ramos.
Substitution, Real Madrid. Isco replaces Toni Kroos.
Substitution, Real Sociedad. Xabi Prieto replaces Juanmi.
Álvaro Morata (Real Madrid) wins a free kick in the defensive half.
Foul by Mikel González (Real Sociedad).
Corner,  Real Madrid. Conceded by Asier Illarramendi.
Corner,  Real Madrid. Conceded by Gerónimo Rulli.
Attempt saved. Mateo Kovacic (Real Madrid) right footed shot from outside the box is saved in the bottom right corner.
Attempt blocked. Sergio Ramos (Real Madrid) right footed shot from outside the box is blocked.
Zaldúa (Real Sociedad) is shown the yellow card for a bad foul.
Gareth Bale (Real Madrid) wins a free kick in the attacking half.
Wales forward Gareth Bale scored twice as Real Madrid opened their La Liga season with a win at Real Sociedad.