Costa scored Atlético's equalising goal in the Copa del Rey final clash against city rivals Real Madrid on 17 May 2013, contributing to the 2–1 triumph – the first in 25 games in a streak stretching back to 1999 – and the tenth win in the tournament, confirmed by Miranda's extra-time header. He and opponent Cristiano Ronaldo had gone into the match as joint top scorers in the tournament, and thus Costa's eighth goal made him the top scorer.
In August 2013, Costa was heavily linked with a move to Liverpool, who allegedly matched his release clause of €25 million and offered him three times his salary at Atlético. Costa, however, chose to stay at the club and renewed his contract until 2018, while also doubling his wages; a few days after this, in the first match of the new season on 19 August, he scored a brace in a 3–1 win at Sevilla.
On 24 September, Costa scored both goals in a 2–1 home triumph over Osasuna to help his team stay level on points with league leaders Barcelona through six games. Four days later, in the Madrid derby, he scored the only goal of the game to record a second win over Real at the Santiago Bernabéu in under five months. For his performances, he was crowned the inaugural La Liga Player of the Month for September 2013. By his 25th birthday on 7 October, he had scored ten goals in eight league matches, equalling his tally from the previous season. All of those matches were won by Atlético, setting a new record for the best start to a season. On 23 November 2013, Costa scored an overhead volley from a cross by Gabi in a win over Getafe; the goal was nominated for the FIFA Puskás Award.
On 22 October 2013, Costa marked his UEFA Champions League debut with two goals against Austria Wien, the first coming after a fine individual effort in an eventual 3–0 group stage away win. On 19 February 2014, in the first knockout round's first leg, he scored the game's only goal at Milan, scoring seven minutes from time after a corner kick from Gabi; he added a further two in the second match, helping Atlético to a 4–1 victory that put them into the quarter-finals for the first time in 17 years.
On 5 March 2013, Costa was called up to the Brazil national team by head coach Luiz Felipe Scolari for friendlies with Italy in Geneva and Russia in London, both taking place late in that month. He made his debut in the first match on 21 March, replacing Fred midway through the second half of the 2–2 draw. Four days later at Stamford Bridge, he replaced Kaká for the last 12 minutes of a 1–1 draw with Russia.
In September 2013, the Royal Spanish Football Federation made an official request to FIFA for permission to call up Costa for the Spain national team. He had been granted Spanish nationality in July. FIFA regulations currently permit players with more than one nationality to represent a second country if, like Costa, he had only represented his first country in friendly matches.
On 29 October 2013, Costa declared that he wished to play international football for Spain, sending a letter to the Brazilian Football Confederation (CBF). Following the news, Scolari commented, "A Brazilian player who refuses to wear the shirt of the Brazilian national team and compete in a World Cup in your country is automatically withdrawn. He is turning his back on a dream of millions, to represent our national team, the five-time champions in a World Cup in Brazil."
Earlier in his Atlético Madrid career, Costa's physical play was used in support of Radamel Falcao, thus ensuring a lower goalscoring rate. After Falcao was sold in 2013, the attack was restructured around Costa by manager Diego Simeone. Simeone, who like Costa was known for his competitiveness and aggression, found ways to enhance his discipline while retaining his determination. In 2014, his club teammate Diego Godín described Costa as the team's "heartbeat," commenting that he "gives us everything," also adding: "Sometimes things aren’t going well and he is able to open up the game with his strength and technique." Nick Dorrington of Bleacher Report described him as a "battering ram of a striker: Strong, quick and tireless in his pursuit of the ball," while the club's manager Simeone lauded his work-rate as being "contagious."