The Tigers, who enjoyed the last of their three Test wins to date against West Indies in July 2009, resumed in Harare needing six wickets to wrap up victory.
Zimbabwe ended day four on 138 for four in pursuit of 401 and reached lunch on 194 for six.
Nightwatchman Shingi Masakadza was first to fall after making an obdurate 24, with Elton Chigumbura following for just two.
The elder Masakadza brother, Hamilton, held his side's innings together and was 73 not out at the interval.
- Sports & Recreation