Gillespie picked up 4-7 in 2.5 overs to send Kenya crashing for 73 all out, the lowest T20 international total, which the Kiwis surpassed easily in the eighth over to begin their campaign on a rousing note.
Peter Fulton ended the one-sided match with two consecutive sixes off Rajesh Bhudia.
In Tuesday's opening match in Johannesburg, West Indian Chris Gayle scored the first century in the shortest version of the game but could not prevent an eight-wicket win by hosts South Africa.
Daniel Vettori celebrated his elevation as New Zealand's Test captain earlier in the day by winning the toss and asking Kenya to bat first at a sun-baked Kingsmead.
Kenya lost the first three wickets without a run on the board, slipped to 4-1 before Collins Obuya and Thomas Odoyo put on 36 for the fifth wicket.
Both batsmen made 18 runs each but it was not enough to help the Africans put up a reasonable total to challenge New Zealand in the group C match.
Vettori, who already captains New Zealand in one-day and Twenty20 cricket, replaced the long-serving Stephen Fleming as Test skipper earlier on Wednesday.
"I am pretty happy the way we bowled," said Vettori. "(Shane) Bond and Gillespie set up the win for us and we now look forward to our other matches."
Fast bowler Bond struck with the first delivery of the match when he bowled Maurice Ouma and then dismissed Tanmay Mishra off his fifth ball.
Gillespie removed David Obuya and captain Steve Tikolo in his first over and returned to polish off the tail to beat England captain Paul Collingwood's previous best figures of 4-22.
Bond, Vettori and Chris Martin took two wickets each as Kenya were shot out in 16.5 overs, falling below Australia's 79 against England at the Rose Bowl in 2005.
New Zealand lost Lou Vincent for 27, but Brendon McCullum (16 not out) and Fulton (21 not out) saw the Kiwis home in style.
The 12-nation tournament features the nine Test nations currently playing Test cricket along with Zimbabwe, Kenya and Scotland.
Kenya next play Sri Lanka on Friday, while New Zealand meet the Sri Lankans on Saturday.