"We have decided to pick Russell as he will be available. There were other candidates but we wanted someone who could be available for us and certainly he has got the experience," said BCB president Nazmul Hasan.
The 44-year-old pipped Mike Hesson and Mickey Arthur in the race to succeed Steve Rhodes, who parted ways with the BCB after the team finished eighth at the 2019 World Cup.
Domingo, who was interviewed by the BCB president Nazmul Hassan in person in Dhaka earlier this month, will take over ahead of Bangladesh's one-off Test against Afghanistan in Chattogram in September.
He is currently the South Africa A coach and will tour India as part of a shadow tour ahead of the senior team's full series in the country in September.
The BCB have already renewed the contracts of batting coach Neil McKenzie and bowling coach Charl Langeveldt, both of whom were part of the support staff under interim coach Khaled Mahmud for the ODI series in Sri Lanka last month. Both McKenzie and Langeveldt were part of South Africa's support staff group during Domingo's tenure as head coach of South Africa from 2013 to 2017.