The 53-year-old former test captain will be supported by assistant coaches Sanjay Bangar (batting), Bharat Arun (bowling) and Ramakrishnan Sridhar (fielding) for the three one-day internationals and one-off test match.
All three retain the roles they held under Zimbabwean Fletcher, whose term as coach came to an end at the conclusion of the World Cup in Australia and New Zealand earlier this year.
India test captain Virat Kohli believes Shastri is the right man for the role.
"He is a guy who does not shy away from responsibility. He is someone who takes the blows on the chin. He keeps moving forward," Kohli told Cricket Monthly in an interview.
"There are no two ways about his thinking. He is the most amazing person to have around this team right now because it is a young unit and the kind of confidence he gives. He has authority; when he speaks people respect him."
Shastri will travel to Bangladesh as "Director of the Indian cricket team", a title he also held for the tour of Australia around the New Year, which culminated in a semi-final defeat in the World Cup.
Another former test skipper, Sourav Ganguly, was among three former India greats appointed to a new BCCI advisory committee on Monday along with Sachin Tendulkar and VVS Laxman.
After the Bangladesh trip, India travel to Africa to play three one-day internationals against Zimbabwe in July.
(Reporting by Nick Mulvenney in Sydney; Editing by John O'Brien)
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