Prosper Utseya (36) and Graeme Cremer (22) added useful runs but Tim Southee, who finished with 3-29, bowled Utseya to close the innings on 162.
Martin Guptill and Brendon McCullum both hit half centuries as New Zealand cantered home with 99 balls to spare.
The Africans chose to bat first in the Group A match at the Sardar Patel Stadium and were quickly in trouble when Hamish Bennett ran out Charles Coventry for a duck with a direct hit in the second over.
Tatenda Taibu (eight) was dropped at cover by New Zealand captain Daniel Vettori but he fell next delivery, trapped leg before by Southee.
Vettori made amends by removing Elton Chigumbura (one) and Regis Chakabva (nought) in his first over as Zimbabwe slumped to 46-5.
Taylor added 40 for the sixth wicket with Greg Lamb before he was adjudged lbw to Scott Styris six short of his half century.
And Greg Lamb followed soon after, run out by Guptill for 18 to leave the score 89-7.
Utseya and Cremer’s resistance took Zimbabwe past 100 but Kyle Mills ended their 33-run partnership with his second wicket of the match before Southee cleaned up Ray Price (11) and Utseya. Guptill began New Zealand’s reply in explosive fashion, hitting two boundaries and a six in the first over.
The duo’s partnership was New Zealand’s highest opening stand in World Cup matches and the highest of any team in this tournament, with Guptill contributing 86 runs and McCullum 76. They shared four sixes and hit 13 boundaries between them as they swatted Zimbabwe’s bowlers to all corners of the pitch.