A product of the Atleti youth system, Dominguez made over 70 appearances for the Spanish side in the past four seasons and was part of the team that won the Europa League in 2010 and 2012, but he had seen his opportunities limited since the arrival of Diego Simeone as manager in January.
The 23-year-old was though still called up to Vicente Del Bosque's preliminary Spanish squad for the European Championship and made his full international debut in the 2-0 win over Serbia last month.
Monchengladbach will initially pay around 8million euros for the player's services with potentially a further 2mil euros to come in add-ons.
"It has been very nice for me to be in this house since I was 13 years old," he told AS.
"I have felt incredible and the last few years we have been able to make Atletico a big club in Europe again. I hope it continues like this and I will continue to feel like an Atleti fan wherever I go.
"I am young, I am ambitious and I want to experience other things and other big leagues. I will always take this club in my heart and I don't have the feeling I am leaving through the back door, I have helped to win two European titles in the past three years.
"I am happy and excited because Germany is a big opportunity for me. I leave privileged to have been a part of this club and wherever I go I will always have these people and this fanbase with me."