The Spain striker - signed from Atletico Madrid last month - gave fans at Stamford Bridge a first-hand glimpse of what they can expect from him in the upcoming season, which Chelsea start at Burnley next Monday.
Having already netted in a friendly at Olimpija, Costa took his pre-season tally to three with a fine brace as he displayed the kind of form that saw him 36 goals for Atleti last season.
Costa's goals came inside the opening seven minutes, allowing Jose Mourinho's men to take their foot off the gas and coast to a comfortable victory.
For Sociedad the result represented a second straight defeat on English shores after succumbing 1-0 to Newcastle United last Sunday, and Jagoba Arrasate will hope his side can turn their form around prior to a UEFA Europa League play-off clash with Krasnodar a week on Thursday.
Prior to the game Costa had promised to make amends for playing a part in Atleti's UEFA Champions League elimination of Chelsea at the semi-final stage last season.
And it took just two minutes for the 25-year-old to show the Chelsea fans that he is capable of acting on that promise, netting with a composed finish into the bottom corner from Ramires' clever pass.
Costa doubled his and Chelsea's tally five minutes later when he capitalised on a rebound from Andre Schurrle's curling effort which crashed against Enaut Zubikarai's left-hand post to slot into an unguarded net.
Chances dried up following that goal but Eden Hazard did come close to finding a third in the 33rd minute as his low free-kick was well kept out by Zubikarai.
The hosts continued to enjoy almost complete control of the game in the second half, and Costa spurned a glorious chance for his treble after 54 minutes as he fired wide of the near post following superb play down the right from Hazard.
A raft of second-half changes slowly disrupted the flow of the game, however, Costa's replacement Fernando Torres came close to making it 3-0, wastefully heading over the crossbar from close range.
But Torres' profligacy mattered little, with Costa's clinical performance giving Mourinho renewed confidence in his foward line despite Dider Drogba's injury and ensuring Chelsea will head to Turf Moor with momentum.
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