Put in to bat after losing the toss, Zimbabwe made 191-8 but Sri Lanka cruised to the target with 14 balls to spare.
Mishara and Kusal Perera put on 117 runs in an unbeaten third-wicket stand as Sri Lanka completed a white-ball double, having won the preceding one-day international series against Zimbabwe 2-0.
Having scored just 35 runs in four previous T20I innings, 24-year-old Mishara made his mark with a match-winning knock, hitting three sixes and six fours from the 43 balls he faced.
"It is a privilege to win a game for my country, I just feel very good," said Mishara.
"There was pressure, of course, but the coaching staff just told me to play my normal game. I just wanted to get into my rhythm and then played my normal game."
Mishara and veteran Perera came together with Sri Lanka 76-2 after 7.1 overs and took control as Sri Lanka put a nightmare 80 all out in the second match on Saturday behind them.
Perera scored as briskly as Mishara, with his unbeaten 46 coming off 26 deliveries. His innings included two sixes and four fours.
Wicketkeeper Tadiwanashe Marumani top scored for Zimbabwe with 51 off 44 balls before being stumped by Mendis off a googly from leg-spinner Dushan Hemantha.
Hemantha (3-38) also took the wickets of Sean Williams (23) and home captain Sikandar Raza (28).