Streak said he will quit the post at the end of the month after deciding not to renew his contract.
"I have decided that I shall leave my role as a national bowling coach when my contract expires at the end the month," Streak said via his agent on Twitter on Friday.
"I feel now is the right time for me and my family to make this decision and I look forward to the future as I look to continue my coaching career," he added.
The Bangladesh Cricket Board (BCB) said Streak had informed them of his decision via email.
"Streak informed us through an email on Thursday that he will not renew his contract," BCB cricket operations chief Akram Khan said on Friday.
Under the Zimbabwean, Bangladesh have not only reached the World Cup knockouts for the first time, but also beat Pakistan, India and South Africa in the same year to qualify for the 2017 Champions Trophy in England.
Streak, who is reportedly eyeing a job at India's National Cricket Academy in Bangalore, is currently the coach of the Gujarat Lions franchise in the Indian Premier League.