Destanoğlu played 18 games at 2018–19 season of TFF U-19 Elite Development League.[3] Following the departure of Loris Karius, Destanoğlu became the first choice goalkeeper in June 2020.[4][5] On 13 June 2020, Destanoğlu made his Süper Lig debut at week 27 encounter of 2019–20 season, which ended 2–1 in favour of Antalyaspor.[6]
He was suspended following a collusion outside the box with opponent's striker Muhammet Demir, on Süper Lig encounter against Gaziantep on 6 November 2020, being the first Beşiktaş goalkeeper shown a red card since Rüştü Reçber's card given in 2007.[10]
On 4 April 2021, Destanoğlu saved a penalty taken Swedish forward Isaac Thelin in a game against Kasımpaşa, held at Recep Tayyip Erdoğan Stadium, ended 1–0 for Kasımpaşa.[11] Because of this, Destanoğlu became the first goalkeeper to save a penalty in a Süper Lig game since 2015–16 when Tolga Zengin saved one.[12]
On 1 October 2022, Destanoğlu has signed a new four-year contract with Beşiktaş, keeping him at the club until 2026[13]