Fleming made 77 not out from 92 balls to help the Kiwis finish on 144 for three and race past Sri Lanka's modest total of 141 with 17 overs to spare.
The tourists were skittled out in just 42 overs as all-rounder Chris Cairns ripped through the middle order to capture four wickets for 33 runs.
Cairns picked up the vital scalps of former captain Sanath Jayasuriya for 43 and top-scorer Tillakaratne Dilshan for 48 to prevent the Sri Lankans from mounting a fightback after they had made a disastrous start to their innings.
After being sent in to bat, Sri Lanka lost four wickets inside the first 13 overs as opener Saman Jayantha (0), captain Marvan Atapattu (4), wicketkeeper Kumar Sangakkara (1) and Mahela Jayawardene (2) all fell cheaply to the bowling combination of Daryl Tuffey, Kyle Mills and Jacob Oram.
Jayasuriya and Dilshan tried to rebuild the innings but both succumbed to Cairns, who completed eight overs despite twisting his ankle when he fell in his first over.
Jayasuriya struck five boundaries and a six from 60 balls before he top edged an attempted hook to Hamish Marshal while Dilshan hit three fours and a six off 79 deliveries when he was brilliantly caught at slip by Fleming.
New Zealand made a shaky start to their reply when Nathan Astle was trapped leg before wicket to Chaminda Vaas for six but the home side quickly resumed control.
The Kiwis lost Mathew Sinclair for 31 and Scott Syris for 12 but were never in danger of losing as Fleming continued to score freely.
The New Zealand captain belted eight boundaries and two towering sixes to post his 38th one-day international and secure the man of the match award.
Sunday's match was the first of five limited-overs internationals between New Zealand and Sri Lanka that will determine second place in the world rankings behind Australia.