Known for his kicking accuracy and game management, Cooney made his senior debut for Leinster in 2011 away to Ospreys before joining Connacht on a short-term loan deal. He made the move permanent ahead of the 2015/16 season before making the switch up to Belfast in 2017, signing a two-year contract at Kingspan Stadium. Cooney made an immediate impression at Ulster, winning Ulster Player of the Year, the Supporters Club Player of the Year and the Rugby Writers' Player of the Year in his debut season. During that season he was also shortlisted for the ECPR player of the season award. His continued excellent form in the 2018-19 and 2020-21 seasons led to him being named in the Guinness Pro14 Dream Team.
In 2022, Cooney won the inaugral URC Difference Maker award for his volunteer work with Belfast Central Mission's befriending service.