Born in the Upper Carniolan town of Škofja Loka, Oblak started playing football at the age of five for his home town club Ločan. At the age of ten, he moved to the Olimpija youth academy, where he remained until the end of the 2004–05 season when the club was dissolved. He then moved to the newly founded Bežigrad, which, after a series of name changes, became Olimpija Ljubljana in 2008. The following year, Oblak rejected a contract from Italian club Empoli in favour of a trial at Fulham, but he eventually did not leave the club and committed his future with a contract extension until 2011. On 17 May 2009, Oblak, aged 16, made his professional debut for Olimpija Ljubljana in the 2008–09 Slovenian Second League in a 7–2 win against Mura 05. Olimpija Ljubljana earned promotion to PrvaLiga for the 2009–10 season season as league winners. On the next season, Oblak was the team’s main goalkeeper, only missing three league games as the club finished fourth in the league table.

Oblak was first called by the Slovenia under-21 team in August 2009, replacing the injured Jan Koprivec. He made his debut 9 September of that year, against France.

On 14 June 2010, Oblak signed a contract with Portuguese club Benfica, who loaned him to fellow top level club Beira-Mar in August 2010. He then joined Olhanense on loan in January 2011, until the end of the 2010–11 season.

Benfica loaned Oblak to another Portuguese team, União de Leiria, for the 2011–12 campaign. He made his league debut on 15 January 2012, in a 2–2 away draw against Nacional.

On 11 September 2012, Oblak made his first appearance for the senior side, starting in a 2–1 away loss against Norway for the 2014 FIFA World Cup qualifiers.

In July 2013, Oblak failed to present himself for pre-season training, claiming he had no contract with Benfica. Late into the following month, he signed a contract extension until 2018, describing the situation as a “misunderstanding.”

After long-time starting goalkeeper Artur made a series of mistakes midway through the 2013–14 campaign, Benfica manager Jorge Jesus dropped him in favour of Oblak, and the latter went on to keep several clean sheets in his first starts, notably in a 2–0 home win against Porto and a 0–0 draw at Juventus in the semi-finals of the UEFA Europa League. He eventually won the league’s Best Goalkeeper of the Year award on 6 July 2014.

On 16 July 2014, Atlético Madrid announced they had reached an agreement with Benfica for the transfer of Oblak, pending a medical examination. Atlético paid €16 million for the Slovenian player, making him the most expensive goalkeeper in La Liga history. Oblak moved to Madrid on a six-year deal as a replacement for Thibaut Courtois, who had returned to play for his parent club Chelsea following his loan expiration. During his presentation, on 22 July 2014, Oblak said, “I don’t come to replace anyone. I come as another player. I’m here along with the rest of the players and goalkeepers. I’ll do everything in my power to defend this shirt and achieve great results this season. I will do everything in my hand to help the team.”

Oblak was an unused substitute in his first competitive fixture on 19 August 2014, the first leg of the 2014 Supercopa de España against Real Madrid, with Miguel Ángel Moyà playing instead. He made his debut on 16 September 2014, in a 3–2 defeat away to Olympiacos in Atlético’s first Champions League group match of the season. His first clean sheet came in his first Copa del Rey match, a 3–0 win away to L’Hospitalet in the first leg of the last 32 on 3 December 2014. On 17 March 2015, he replaced the injured Moyà in the 23rd minute of a Champions League last 16 second leg against Bayer Leverkusen, and kept a clean sheet in a 1–0 home victory. The tie went to a penalty shootout, in which he saved Leverkusen’s first attempt by Hakan Çalhanoğlu in an eventual triumph. Four days later, due to the injury, he made his league debut, keeping a clean sheet in a 2–0 home win over neighbours Getafe.

In February 2016, Oblak was rewarded with a contract extension through 2021 with a reported release clause of £77.5 million. On 3 May 2016, Oblak saved Thomas Müller’s penalty at the Allianz Arena in the second leg of the Champions League semi-finals; although Atlético lost the match 2–1, they advanced to the final on away goals. He was included in the squad of the season of the 2015–16 Champions League. As the domestic season ended, he won the Ricardo Zamora Trophy for best goalkeeper, having conceded 18 goals in 38 games, equalling the 22-year-old record of Deportivo de La Coruña’s Francisco Liaño.

Oblak received attention on 15 March 2017, in a Champions League game against Bayer Leverkusen, where he made three saves in succession to ensure a goalless draw and passage to the quarter-finals. He told UEFA’s website “These are things that happen. Sometimes you save three efforts and other times they score all three.”

On January 28 2018, Oblak played his 100th league game for Atlético when he started the game against Las Palmas. In the 100 league games, he kept 59 clean sheets and conceded only 54 goals. In the 2018 UEFA Europa League Final, Oblak kept a clean sheet as his side won 3–0 against Olympique de Marseille, winning their third Europa League title in eight years. This was Oblak’s first European title, having been a runner up twice before.

On 6 November 2018, Oblak kept a clean sheet in a 2–0 home win over Borussia Dortmund in the group stages of the UEFA Champions League. By doing so, he kept his 100th clean sheet for Atlético, a feat which he achieved in 178 official games. On April 17 2019, Oblak agreed to a four-year contract extension with the club, keeping him at the club until 2023.

Ahead of the 2019–20 season, he was named as vice-captain of the team, behind Koke. On 21 October 2019, Oblak was nominated for the inaugural edition of Yashin award, the Ballon D’Or of goalkeepers. He finished in fourth place. On 6 December 2019, Oblak kept a clean sheet in a 0–0 away draw against Villarreal. By doing so, he broke the club record of Abel Resino for most clean sheets with his 96th clean sheet in his 169th league game.

On 6 September 2019, Oblak captained Slovenia for the first time.

On 11 March 2020, following Atlético Madrid’s 3–2 away victory over defending champions Liverpool in the last 16 of the UEFA Champions League, Oblak, playing his 50th Champions League game, was awarded the Man of the Match award by UEFA, after making nine saves to help his team progress to the quarter-finals with a 4–2 aggregate victory. On 17 June 2020, Oblak kept a clean sheet in a 5–0 away win against Osasuna, his 182nd league game, and by doing so, became the fastest goalkeeper to achieve 100 clean sheets landmark in La Liga history, breaking an almost 50-year old record of Miguel Reina who did the same in 222 league appearances. Oblak also became the first non-Spanish goalkeeper to keep 100 clean sheets in La Liga history. On 27 June 2020, Oblak played his 250th game for Atlético as his side beat Deportivo Alavés 2–1. He has reached the milestone just one week after equalling José Francisco Molina as the second goalkeeper with the most appearances in the club’s history.