Shastri will replace former captain Kumble, who stepped down as coach last month after falling out with captain Virat Kohli despite enjoying a successful run during his year in charge.
The Indian cricket board (BCCI) also named former India seamer Zaheer Khan as the team's bowling consultant, while former batting great Rahul Dravid was appointed as batting consultant for overseas test series.
Shastri, also a renowned TV commentator, was appointed India's team director in August 2014 with Duncan Fletcher as head coach.
Fletcher's term came to an end following India's semi-final defeat at the 2015 World Cup in Australia and New Zealand.
Last year the BCCI formed the Cricket Advisory Committee (CAC), comprising former players Sachin Tendulkar, Sourav Ganguly and VVS Laxman, who chose Kumble for the head coach position ahead of Shastri, who had also applied.
"Shastri has been the team director in the past, and brings with him a wealth of experience, both as player and as coach," the BCCI said on Tuesday.
"Khan is a contemporary cricketer who assumed the mentoring role to a young fast bowling unit even during his playing days with India. His inputs will be of immense value to our young fast bowlers, who have shown tremendous potential in the last couple of years.
"As the Indian team is to play test cricket overseas in the next two years, Rahul Dravid’s expertise as batting consultant will be immensely helpful for youngsters."
Kumble's original contract expired with India's Champions Trophy campaign, but the former leg-spinner had been tipped to get an extension after leading the country to 12 victories in 17 tests with one loss.
The BCCI invited applications for the role last month and Kumble, India's most successful bowler, re-applied. He later stepped down saying his partnership with Kohli was "untenable".
Shastri had not initially applied for the position but put in his name once Kumble was out of the equation.
He was among five candidates interviewed by the CAC on Monday to fill the vacant position, along with Indians Virender Sehwag and Lalchand Rajput, Australian Tom Moody and Englishman Richard Pybus.
The first challenge for the new coaching set-up will begin later this month when India tour Sri Lanka to play three tests, five one-dayers and one Twenty20 international.
(Reporting by Sudipto Ganguly; Editing by Ed Osmond and Ken Ferris)
