Mendis, 27, marked his international return after an eight-month injury lay-off with the devastating haul to wreck Zimbabwe for 100 in 17.3 overs in the Group C match in Hambantota.
Zimbabwe could never cope with Mendis's unreadable deliveries and they collapsed like a pack of cards in pursuit of the daunting target of 183 set by Sri Lanka at Mahinda Rajapaksa Stadium.
Sri Lanka, who lost the toss and were sent into bat, owed their total of 182 for four to Kumar Sangakkara (44) and Jeevan Mendis (43 not out), who shared a 94-run stand for the fourth wicket.
"It's tough playing a team like Zimbabwe," said Mahela Jayawardene, the Sri Lanka captain.
"You can't take anything for granted. We had to make sure we played to our potential. It's a good wicket, just needed to get used to it. Our guys batted really well, and the bowlers then did a super job."
Ajantha Mendis struck two blows in his first over, removing Vusi Sibanda (11) and Zimbabwe captain Brendan Taylor (0) off successive deliveries. He next had Zimbabwe's top-scorer Hamilton Masakadza (20).
He returned for his second spell and took four more wickets to end any lingering Zimbabwean hopes – and improve his own international Twenty20 record figures of six for 16, set against Australia at Pallekele last year.
Jeevan Mendis took three for 24 to cap a great all-round performance.
"We are a better side than what we showed," said Zimbabwe's captain Taylor. "Credit goes to Sri Lanka. They showed why they are one of the favourites to win the tournament. We have played enough cricket to go back from today, rethink and come back."
Sangakkara, the newly crowned ICC player of the year, hit a 26-ball 44, studded with a six and two boundaries, while Jeevan Mendis smashed four boundaries and a six during his unbeaten 43 off 30 balls.
Both the left-handers lifted Sri Lanka from 82 for three after they were put into bat.
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