[1][2] On 5 July 1996, Costa was appointed bishop of the Roman Catholic Diocese of Dinajpur and consecrated on 6 September 1996 by Adriano Bernardini.
On 6 April 2011 he was appointed bishop of the Roman Catholic Archdiocese of Chittagong and was installed on 27 May 2011.
He was appointed Archbishop of the Roman Catholic Archdiocese of Chittagong on 2 February 2017 by Pope Francis.
[3][1][2] On 13 June 2020, Costa was admitted to hospital after resulting positive to COVID-19, but later recovered from the disease.
On 9 July 2020, Costa suffered a stroke, and was put on life support two days later.