In a dramatic finish, Worcester paceman Kabir Ali grabbed the last two Zimbabwe wickets with the scores level.
The tourists had been set 176 to win and at 128 for eight the match looked over.
Streak had not been expected to feature after limping off on the first day with a back spasm. He did not bat in Zimbabwe's first innings but came in at number nine on Monday and using a runner managed to make 37.
With the scores level and Zimbabwe on 175 for eight, he was caught in the slips off the bowling of Ali, who then bowled Doug Hondo for a duck two balls later. Ali ended with figures of five for 48.
Zimbabwe's first Test against England starts at Lord's on May 22. The second and final match of the series is at Durham's ground in Chester-le-Street from June 5.
