The Spaniard marked his home debut with a goal in the opening minute after running onto Ramires' pass and finishing into the bottom corner from a tight angle.
Costa continued his impressive pre-season form with his second six minutes later after tapping in a rebound after Andre Schurrle's curling effort rattled the post.
Chelsea were rampant in the early stages with Ramires' shot deflected wide with the goal at his mercy and Eden Hazard forcing the keeper into a fine save from a free-kick.
Sociedad were poor and their only effort of the first period came from Esteban Granero's curling effort that flew wide.
Normal traffic resumed in the second half with Costa firing wide from close range and his eventual replacement Fernando Torres heading over from six yards.
Gonzalo Castro forced Thibaut Courtois into a save at his near post before flashing a drive wide in the closing stages as Chelsea held on for a comfortable victory in their first friendly at Stamford Bridge in eight years.
2' - GOAL! - Ramires feeds the ball to Costa, who slots the ball into the corner. What a start!
3' - RAMIRES MISS - Good long ball from Costa to Ivanovic who cuts the ball back to Ramires, whose deflected shot goes wide with the goal at his mercy.
6' - GOAL! Schurrle hits the post with a curling effort and Costa taps in the rebound from four yards, 2-0 Chelsea.
18' - GRANERO EFFORT - Granero sends a curling effort from 20 yards inches wide, decent effort.
33' - HAZARD FREE-KICK - Hazard will take the free-kick and fires it low to the keeper's right which he parries, good save.
53' - COSTA CHANCE - Hazard on the right cuts the ball back to Costa who fires his shot inches wide at the near post. Great ball from Willian to set up the move.
63' - PENALTY APPEAL - Fabregas appeals for handball with a shot from long-range, it looked to have struck the defender's forearm.
72' - TORRES MISS - Torres ghosts in between two defenders and puts a header over from six yards. Good movement, shame about the finish.
77' - COURTOIS SAVE - A counter attack from the visitors but Courtois is down quickly to save Castro's effort.
80' - CASTRO SHOT - Substitute Castro flashes an effort across goal with power, decent effort from a tight angle.
88' - GREAT SAVE - Salah's shot is blocked before Zubikarai tips away Oscar's shot, great save.
TALKING POINT: Will Thibaut Courtois be Chelsea's first-choice keeper? The Belgian started tonight and looked solid in commanding his box. The Blues' line-up tonight looked to be the strongest team available so Jose Mourinho could opt for youth over experience in the opener against Burnley. Peter Cech will feel hard done by if that's the case having kept 11 clean sheets last season.
MAN OF THE MATCH
Diego Costa (Chelsea): The striker seems to have settled into the club without any problems. He took his first goal well from a tight angle before showing his goal poaching instincts for the second. The Spaniard also linked up play well and used his strength when he dropped deep in Didier Drogba fashion.
Chelsea: Courtois 6, Ivanovic 6, Cahill 5, Terry (c) 5, Ake 6, Matic 5, Fabregas 7, Ramires 6, Hazard 7, Schurrle 7, Costa 9.
Subs: Azpilicueta 6, Christensen 5, Filipe Luis 6, Zouma 5, Mikel 5, Oscar 5, Salah 5, Willian 7, Torres 5.
Real Sociedad: Zubikarai 6, Zaldua 5, Ansotegi 4, Iñigo Martínez 6, De la Bella 5, Markel 5, R.Pardo 6, Granero 7, X.Prieto 5, Canales 6, Agirretxe 4.
Subs: Yuri 4, C. Martínez 5, Estrada 6, Elustondo 5, A.Oyarzun 5, I.Hernández 5, Castro 6, Gonzalez 5, Hervias 5.
- Sports & Recreation
- Diego Costa
- Real Sociedad
- Thibaut Courtois
- Eden Hazard
- Esteban Granero
- Fernando Torres