The Blues made the perfect start when Costa opened the scoring in the second minute.
The Spain striker was unleashed by a fine Ramires through-ball and made no mistake in dispatching past Sociedad goalkeeper Enaut Zubikarai from a tight angle to give the hosts a 1-0 lead.
Costa then doubled Chelsea's lead five minutes later when he tapped home from close range after Andre Schurrle's shot from the edge of the area crashed off a post and into his team-mate's path.
Costa missed a chance to snatch a hat-trick in the second half while Fernando Torres also fluffed a clear-cut opening, while Thibaut Courtois - largely untroubled for most of the match - had to be alert to deny Chory Castro in the 77th minute.
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