Together with Hamish Marshall, who made 50 not out, he shared an unbroken second wicket stand of 134 at the Sir Vivian Richards Stadium.
Marshall's six off Mohammad Ashraful saw New Zealand win with more than 20 overs to spare.
Fleming's 90-ball innings, which featured three sixes and 10 fours, was only his eighth century in 275 one-day internationals.
Marshall faced 54 balls with one six and two fours.
Victory left New Zealand, who carried forward two points for a win over fellow Group C qualifiers England, level at the top of the standings on six points with world champions Australia.
The Black Caps' second win in the Super Eights, after a seven-wicket defeat of hosts West Indies on Thursday, was set up by a fine bowling display that saw Bangladesh bundled out for 174.
Jacob Oram (three for 30) made the initial breakthrough with two top order wickets and a run out before wrapping up the innings with nine balls to spare by bowling last man Syed Rasal for 10.
Fellow all-rounder Scott Styris followed up with four for 43.
The only hiccup when New Zealand batted saw makeshift opener Peter Fulton (15) chip Rasel to Tamim Iqbal at mid-on.
All the Bangladesh top order made it past 20 but no-one was able to press on to a major score.
However, tailender Mohammad Rafique struck two sixes in his unbeaten 30 and put on 34 for the tenth wicket with Rasel.
Bangladesh, with opener Javed Omar replacing Shahriar Nafees following the 10-wicket defeat against world champions Australia, made steady progress.
Javed, who'd shared a stand of 79 with Iqbal during Bangladesh's two-wicket warm-up victory against New Zealand, showed his class by lofting left-arm quick James Franklin for four over mid-off.
But an opening stand worth 55 ended in the 17th over.
Teenage left-hander Iqbal, on 29, tried to sweep Oram but over-balanced and was stumped by Brendon McCullum.
Omar went soon afterwards for 22, cutting a ball from pace bowler Oram too close to him for the stroke and edging through to wicket-keeper McCullum.
The batsman walked off without waiting for the umpire's decision and Bangladesh were 62 for two.
Aftab Ahmed and Hasan rebuilt with a third-wicket partnership of 43 although the run-rate remained at under four an over.
Ahmed, looking to up the tempo on 27, tried to attack medium-pacer Styris but holed out to substitute fielder Mark Gillespie at long-on.
Bangladesh captain Habibul Bashar was then run out for nine, going for a second by Oram's direct hit from the point boundary.
And 122 for four was transformed into 127 for five when Saqibul Hasan (25) was beaten for pace by Shane Bond, who took an impressive two for 15 off 10 overs, and clean bowled.
That became 127 for six when new batsman Mushfiqur Rahim went for a five-ball nought, superbly bowled by a full length Bond delivery that moved away late.
Styris struck twice in three balls to remove Mashrafe Mortaza and Abdur Razzak but left-hander Rafique hit back by hoisting the medium-pacer high over long-on for six.
Bangladesh next play South Africa in Guyana on Saturday with New Zealand taking on Ireland there Monday.