Sunday saw debut defeats in Spain for Villarreal and Getafe, both of whom were expected to do well this season, dpa reported.
Villarreal crashed 1-0 away to newly-promoted Real Sociedad while Getafe lost 3-1 against Espanyol.
Real Sociedad - who have just returned to La Liga after three years in the second division - dominated from start to finish and won thanks to a 58th minute goal from Xabi Prieto, who finished calmly after playing a clever one-two with Joseba Llorente.
Llorente said that "we knew that it was important for us to make a good start today and that is exactly what we have done, thanks to our hard work."
Getafe, who finished sixth last season, never looked comfortable against lively Espanyol, who had a two-goal hero in Argentine striker Pablo Osvaldo.
Osvaldo gave the hosts the lead in the 34th minute after being set up by Jose Callejon and then took advantage of a blunder by Miguel Torres to make it 2-0 10 minutes into the second half.
Espanyol defender Ernesto Galan scored a clumsy own goal, but Jesu Datolo made it 3-1 in stoppage time after another swift break from Callejon.
Sunday's other early games were 0-0 draws between Deportivo Coruna and Zaragoza, and between Osasuna and Almeria, for whom defender Carlos Garcia was sent off.
Later on Sunday, defending champions Barcelona kick off away to Racing Santander, then Real Madrid start their campaign away to Mallorca.