Mourinho splashed out £32 million ($53 million) to lure Costa from Spanish champions Atletico Madrid and the Blues boss is convinced even that hefty price tag will look cheap by the end of the season.
The 25-year-old certainly looks capable of justifying Mourinho's faith after making a confident bow for his new club at Stamford Bridge.
He took just 57 seconds to open the scoring with a cool finish from an acute angle and bagged his second with a tap-in after Andre Schurrle's shot hit a post in the seventh minute.
As well as his goals, Mourinho was also impressed with the way Costa linked with attacking midfield trio Schurrle, Eden Hazard and Cesc Fabregas, while working tirelessly to close down Sociedad when they were in possession.
His qualities look ideally suited to the Premier League and Mourinho said: "Diego is a team player, he works a lot, he is not just about goals. He is about defensive work, his mentality, his ambition.
"This is why we didn't buy a striker in the last transfer window - we were waiting for him.
"He looks very adapted to his team-mates and the style of play.
"I won't say he is adapted to the Premier League yet because it is a very difficult job to be a striker in this country, but hopefully he does well.
"You can consider that it is a bargain because the market is crazy.
"The values for strikers are very high, so for a young striker like him, a champion in Spain and a Champions League finalist, you can say the price Chelsea paid is very much in our favour."
- £80 million overhaul -
After finishing last season without a trophy, Mourinho has carried out an £80 million overhaul of his squad, with Costa, Fabregas and Filipe Luis the headline signings.
The Portuguese coach is delighted with his new acquisitions and, although he refused to bill Chelsea as title favourites, he predicted great things for the club in the long-term.
"I think we bought three players in Filipe, Diego and Fabregas in the best stage for a football player between 25 and 30.
"They are three experienced players with good qualities to improve immediately our team.
"They are ready so normally our team will improve, the squad has improved for sure.
"We showed some qualities today that in the past we didn't.
"No manager has a perfect squad but we like very much our squad, we don't say they are special, but it is a squad that has big possibilities for the next five years."
Belgian youngster Courtois, so impressive on loan at Atletico last season, started against Sociedad and Mourinho says he has chosen his number one for the opening day.
"Courtois played 90 minutes because he was injured and didn't play last week," Mourinho said.
"The decision that he played today is not me saying he is the first choice for Monday.
"I know who will play. The players don't know. They have to know first. I will tell them on Sunday."
Mourinho also revealed that Blues legend Didier Drogba, back from Galatasaray for a second tour of duty at the Bridge, could still recover from his ankle injury in time to feature against Burnley.
"Didier is good, I don't want to say ready for Monday, but with the quality of our medical department and the mentally he has, I'm not saying it is impossible for him to recover for Monday either," he added.
- Sports & Recreation
- Jose Mourinho
- Diego Costa
- Premier League
- Real Sociedad
- Cesc Fabregas