Cookson was preferred to McQuaid in a vote that saw him win 24-18 and end the Irishman's eight-year spell in the job.
"It is a huge honour to have been elected president of the UCI by my peers and I would like to thank them for the trust they have placed in me today," Sky Sports News quotes him as saying.
"My first priorities as president will be to make anti-doping procedures in cycling fully independent, sit together with key stakeholders in the sport and work with WADA to ensure a swift investigation into cycling's doping culture.
"Finally, while there have been some difficult moments between myself and my opponent Pat McQuaid during this election contest, I would like to thank Pat for the contribution he has made to cycling during his long career. I wish him well in whatever he goes on to do."
Cookson was awarded an OBE in 2008.