The Brazil-born attacker arrived at Stamford Bridge in July after scoring 27 goals in 35 Liga games for Atletico Madrid last season, helping Diego Simeone's side win the Spanish title and reached the Champions League final.
Costa netted twice on his home debut for Chelsea as the club rounded off their pre-season preparations with a 2-0 win over Real Sociedad, adding to earlier strikes against Olimpija Ljubljana and Fenerbahce, and Mourinho is confident the big-money move will turn out to be a bargain.
"What can we expect? I expect Diego and Diego is a team player who works hard, he’s not just about goals," Mourinho told the club's official website. "It’s about his defensive work, mentality and ambition.
"This is the player we bought and we were waiting for since last season. This is why we didn’t buy a striker in the winter market. We were waiting for him, we have him and hopefully everything goes well.
"He looks adapted to the team, and the Premier League, not just Chelsea, should be happy. Sometimes we lose important players going to other countries and sometimes we bring new top players here.
"You can consider that [a bargain] because the market is crazy and the values are very high. For a striker like him, a champion in Spain, a Champions League finalist, a 25-year-old, we consider the price was very much in our favour."
Chelsea kick off their Premier League campaign on Monday night against newly-promoted Burnley, and Mourinho is happy with the "strong nucleus" of his team after a successful summer business; bringing in Costa, Luis Filipe and Cesc Fabregas.
"We bought three players, in Filipe, Diego and Cesc, at the best stage for a player, between 25 and 30," he continued. "They are three experienced players with qualities to immediately improve our team.
"Last season we went to a different profile when we bought people like [Mohamed] Salah and [Andre] Schurrle, who needed time.
"Obviously we also went for [Kurt] Zouma but the strong nucleus of Filipe, Diego and Cesc is ready.
"The squad has improved for sure. This is the Premier League, every team tries to be better. Are we ready for that? Yes, but this is England.
"In some countries only two teams can win the league, sometimes only one, but here it’s five or six.
"The only thing I can say is that I very much like the squad. The club were fantastic because they gave me the three targets I asked for.
"We have the squad we want to have, it’s a squad for tomorrow, for next season and also with big possibilities for the next five to 10 years."