Mendis, recently named the ICC's Emerging Player of the Year, finished with 3-15 from three overs after ripping the heart out of the Zimbabwe middle order including dismissing Prosper Utseya with his 'carrom ball', which turns away from the right-handers.
Tatenda Taibu, the former captain, finished on 45 to push Zimbabwe to 106-8 off 17 overs in match shortened by the damp outfield at the King's City venue.
In reply, Sri Lanka openers Tillakaratne Dilshan and Mahela Udawatte put on 36 in four overs, before Udawatte was caught behind.
Left-arm spinner Ray Price bowled an economical spell which yielded two wickets including debutant Jeevantha Kulatunga and Dilshan.
Thilina Kandamby and skipper Mahela Jayawardene were both run out as Zimbabwe reduced the Asian side to 77-5 before Chamara Kapugedera and Farveez Maharoof saw their side home.
Pakistan were due to face Canada in the second match of the four-nation tournament later Friday.