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.