A brace from Costa saw Diego Simeone’s side record a 4-2 win against Champions League holders Madrid in Tallinn on Wednesday.
Speaking after the game, the Spanish international said: “With the desire and the ‘cojones’ that we have, we can fight against any team.
“It was a victory that we had been looking for a long time, I am very happy.
“They won two finals against us, we wanted to win a European final against them. It was very important to get confidence ahead of the season.”
Antoine Griezmann insists Atletico Madrid winning the Super Cup justifies his decision to extend his stay at the Estadio Wanda Metropolitano.
The 27-year-old was a key target for La Liga champions Barcelona, but announced he would stay at Atletico during the World Cup.
The French international produced a mixed performance in Wednesday’s Super Cup win in Tallinn, replaced after 57 minutes as he continues to build up his fitness levels ahead of the new season.
Diego Costa’s superb brace and goals in extra time from Saul Niguez and Koke saw Los Rojiblancos overcome their Madrid rivals.
“I stayed because there was a good project,” Griezmann said to Movistar.
“I have confidence in this club, in Diego (Simeone)… Today I saw that I was not wrong.
“I was a running in water a bit because it was my second game.
“In the end the important thing was to get ahead on the scoreboard, we made a very complete final.
“We know that physically the two teams have not reached 100 per cent, the key was the desire to win.
“It was time for us to win the UEFA Super Cup again, it was very important to start the season like this.”
Julen Lopetegui stressed Real Madrid “have to improve in all facets” after their first competitive match under his management saw them suffer a 4-2 extra-time defeat to bitter rivals Atletico Madrid in the UEFA Super Cup.
Atleti’s Saul Niguez and Koke struck in quick succession in the additional period at Tallinn’s A. Le Coq Arena following a 2-2 draw in normal time.
Real had been 2-1 up, with Karim Benzema cancelling out Diego Costa’s first-minute opener and Sergio Ramos then converting a 63rd-minute penalty, before Costa netted his second of the contest to draw things level.
Lopetegui told AS in his post-match press conference: “We had good game phases but it was not enough.
“We have to congratulate Atletico, I believe that they have won deservedly.
“We are not happy and the negative is that we have conceded four goals and this is not normal. This will have to be improved, without a doubt.
“We have to improve in all facets, we do not like when we make mistakes and we have to improve in everything, not only in individual errors.
“(It is a) feeling of frustration and sadness. But knowing that responsibility forces us to get up and try to improve and start the league as we want to start.”
Real, having not only seen Lopetegui’s predecessor Zinedine Zidane but also club record goalscorer Cristiano Ronaldo depart the club over the summer, begin their 2018-19 La Liga campaign on Sunday when they host Getafe.
While they have won the Champions League in each of the last three years and four of the last five, they have claimed the La Liga title only once in that time, and finished third last term.
Lopetegui said when asked about Real’s hunger having won the Champions League so many times: “We want to focus on the season without thinking about the last years and get the best out of the team.
“We want to get up now and think about the league, which makes us excited. The team has won few La Ligas in recent seasons and we want to start well.
“They are not less hungry for having won the Champions League. They were hungry, eager, excited to win this title.”