Back in the top-flight after a two-year absence, Betis were third with six points from two games behind Real and Valencia on goal difference, the only sides left with 100 percent records.
Champions Barcelona are fourth with four points after a 2-2 draw at Real Sociedad on Saturday.
"We have scored two goals, we have six points and we haven't conceded yet. This is the way forward," Betis coach Pepe Mel (pictured) told reporters.
The Spanish league's timetables for the new campaign have been altered to avoid matches being played at the same time, a decision which has met with criticism from many quarters, especially over the new midday slot.
"I have been trying to see the person responsible for these times," Mel added. "Because one day something unfortunate will happen."
Espanyol's pony-tailed striker Sergio Garcia, a member of Spain's victorious Euro 2008 squad, headed the opener and showed quick thinking to hook in the winner in the 73rd minute, after Fernando Llorente had nodded Bilbao level in the 63rd.
Bilbao's new coach, Argentine Marcelo Bielsa, is still searching for his first win after compatriot Cristian Alvarez pulled off some sensational saves in the Espanyol goal.
Osasuna's Nino and Alvaro Cejudo with a thumping volley scored within two minutes of each other in the first half, before David Barral pulled one back for Sporting in Pamplona.
