Man-of-the-match Vaas took two wickets in his first two overs to reduce New Zealand to four for two. They never recovered and ultimately lost their first game of the tournament by six wickets.
Equally adept on any surface in any country, Vaas first accounted for Fleming with the fifth ball of his first over.
It was the 11th time in one-day internationals that he has dismissed the Kiwi skipper who averages just over 10 against him.
He followed up by dismissing Ross Taylor caught behind, also for a duck, and completed his day with the wicket of New Zealand's other opener Peter Fulton caught for 28.
Vaas's immaculate performance on a slow, low pitch was the highlight of a fine all-round Sri Lanka bowling display.
'CONTROL THINGS'
Muttiah Muralitharan joined the injured Malinga and Australian Glenn McGrath on top of the wicket-taking table with 15 and part-time off-spinner Tillakaratne Dilshan took one for 44 from his eight overs.
Sanath Jayasuriya was economical as usual, conceding 44 from his 10 overs.
"After losing the toss we managed to control things," captain Mahela Jayawardene told a news conference.
"We are not a one-man team. Over the last 12 to 16 months we have had injury problems. What we do is adapt to whatever combination that we play.
"Vaasy put his hand up and really performed and so did Murali."
Jayawardene said Malinga, who has torn a ligament in his left ankle, could possibly have played on Thursday.
"We didn't want to take the risk," he said. "We wanted to make sure he is 100 percent."
Sri Lanka next play unbeaten champions Australia on Monday in Grenada.
The seven-week World Cup culminates in the final in Bridgetown, Barbados on April 28.
