Costa, a £32million acquisition from Atletico Madrid in July, scored twice in the opening seven minutes as Jose Mourinho's side geared up for their Premier League opener at Burnley on Monday with a deserved victory.
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.