Bilbao, who were crushed 4-1 by Barca in the King's Cup final last May, took the lead in the traditional curtain-raiser to the new season, through Oscar de Marcos at the end of the first half.
Pedro set up Xavi for the equaliser early in the second half, with a cut back in the area, and then scored a stunning second from Xavi's lay-off soon after.
The 22-year-old forward, another product of the Barca youth system, jinked past a defender and let fly with a fierce drive from outside the area.
"I am very pleased with Pedro," Guardiola said after the game. "I am sure he will play a big role in the side this season. He worked very hard and is very enthusiastic, wherever you play him."
Barca were missing Lionel Messi, Andres Iniesta and new signing Zlatan Ibrahimovic but dominated possession.
Home keeper Gorka Iraizoz tipped Bojan Krkic's long-range effort on to the bar and pushed another from Daniel Alves over the top as Barcelona pressed early on.
Xavi blazed over from close range after a slick passing move in the 24th minute, but Bilbao's pressuring in midfield started to force some errors from the visitors.
Bilbao youngster De Marcos nearly caught out Victor Valdes with a cheeky long-range lob as halftime approached, but had more luck when his shot deflected off Carles Puyol and over a stranded Valdes for the opener.
But Xavi and then Pedro demoralised Bilbao with their goals in the second half, putting Barca in control for the return leg at the Nou Camp next Sunday.
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