No other members of the constitutional body had filed nominations for the post.
Chakma, a three-time member of the tribal council, had contested the council polls in April 2018 on an MNF ticket from the Parva constituency.
He has replaced H Amaresh Chakma, who was removed from the post through a no-confidence motion on March 26 during the recently-concluded budget session.
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Deputy chairman of the council Mohesh Boran Chakma also submitted his resignation soon after the election of the new chief, the official said.
Sources in the MNF said he stepped down from the post "voluntarily".
The CADC was created in 1972 for the Chakma tribals living in the southwestern part of Mizoram.
It has 20 elected members and four nominated members.
Currently, the MNF has a strength of 18 members, while the BJP has only two.
Durjya Dhan Chakma of the ruling MNF was on April 5 appointed as the Chief Executive Member of the CDAC, following the resignation of Rasik Mohan Chakma on March 29.