They followed the Sri Lanka score of 171 for seven in the allocated 28 overs following morning rain with 166 for seven to lose by only five runs.
This gave the tourists a 3-0 series victory at Harare Sports Club.
With 10 runs to get in the last over Zimbabwe managed only four. After much debate the Sri Lankans decided that Muttiah Muralitharan should be called upon to prevent it, and he duly succeeded.
Zimbabwe's sudden emergence from the depths of batting failure was highlighted by their opening batsmen Hamilton Masakadza and Vusi Sibanda passing the 67 runs innings total Zimbabwe made on the second ODI on Saturday.
And they lost only one wicket in going past 124, their total in the first ODI on Thursday.
Masakadza was their top scorer with 77 in 71 balls (seven fours and two sixes).
Sri Lanka's total after being put into bat was modest.
Kumar Sangakkara top scored for the second time and he recorded the first half-century of the tour. He has now reached a career total of 6,649 runs in ODI's.
Tawanda Mupariwa took four for 39 in six overs and Elton Chigumbura three for 37 in five.
The tourists must have expected this total to be more than sufficient, but the Masakadza-Sibanda combination slammed 33 runs off Farveez Maharoof's first four overs and they continued to speed along at well over the required runs schedule.
The first wicket fell for 76 in the 12th over and their second on 134. They required 38 runs to win in 36 balls and were well placed to do so. But Muralitharan put the brakes on with four for 21 in six overs to deprive the Zimbabweans of victory.
This economic spell was the main contributory factor for the Sri Lankans. The secondary one was Tatenda Taibu's insistence of reverse sweeping, which brought only a scattering of singles.
The fourth ODI is at this ground Friday, and even though the series is settled, Zimbabwe's near miss should ensure heightened interest.
With the series secured Sri Lanka are free to experiment. They have played the same eleven in three matches. But Thilina Kandamby, Dilhara Fernando and Dhammika have yet to take the field.