The oath, administered by President Md Abdul Hamid, was held at the Bangabhaban at 11am.
Prime Minister Sheikh Hasina, Speaker Shirin Sharmin Chaudhury, retired Chief Justice Md Muzammel Hossain and other previous chief justices were present at the ceremony.
The government appointed Justice Sinha to the office last Monday.
Justice Sinha, the most senior Appellate Judge in the Supreme Court, succeeds Justice Md Muzammel Hossain who retired on Jan 16.
He is the 21st head of judiciary in Bangladesh and the first non-Muslim to hold the top position in this Muslim-majority country.
He hails from the minority Bishnupriya Manipuri community.
Justice Sinha was on the Appellate Bench that heard the 13th Amendment appeal and scrapped the provision for caretaker government to oversee general elections.
He was also part of the bench that heard the 2009 appeals in the Bangabandhu Murder Case and has been an appeals judge in the ongoing war crimes trials.
He will be in office for more than three years with his retirement due in early 2018.