Streak resigned after he allegedly made several demands including the team selection criteria and the immediate review of the selection panel, which the ZCU said they could not meet.
"Streak raised a number of issues and demanded that they be met by Monday or else he would resign from all forms of cricket," said the ZCU.
"Instead of accepting his demands, the ZCU Board unanimously accepted his resignation from all forms of cricket with immediate effect.
Taibu, 20, becomes the new captain and will take charge of the team for the first time when Zimbabwe plays against against Sri Lanka in two weeks time.
Streak had earlier on Friday presented the cricket union's board of directors with an ultimatum during their quarterly meeting.
He had wanted the selection panel to be reduced from five to four, all of whom should have played first class or Test cricket. "None of the black directors currently running the game in the country meet these criteria," said ZCU in a statement.
"Streak said he had no confidence in the teams being selected and was under stress as he was doubling up as the players' representative since the players were continuously presenting grievances to him," the cricket body said.
ZCU said earlier in the week Streak had expressed lack of confidence in the team's coach Geoff Marsh accusing him of misrepresenting him to the selectors. "The directors expressed anger at the move taken by Streak saying that one of the conditions of his assuming the post was that he would captain any side selected," said ZCU.
The directors also considered Zimbabwe's poor performance against Bangladesh early this year.
Streak's resignation comes as two International Cricket Council chiefs concluded their visit to Harare on Friday as the debate over England's proposed tour later this year rumbles on.
Streak, who has played 59 Tests and 183 One-dayers, was recently signed by County outfit Warwickshire for the coming season.
He made his international debut against Pakistan in 1993-94 and has career Test wickets of 202 and best bowling figures of 6-87. He also made 1,814 Test runs with a top score of 127 not out.
Streak also took 234 One-day wickets with a best of 5-32 and hit 2,752 runs with a highest score of 79 not out.