New Zealand completed a clean sweep of wins in World Cup Pool A - but they were given one of their toughest tests by Bangladesh.
Mahmudullah's second successive century led Bangladesh to 288 for seven in their 50 overs, with Soumya Sarkar adding 51 in a third-wicket stand of 90 and Sabbir Rahman crashing a quick 40 from number seven.
New Zealand had a centurion of their own in opener Martin Guptill, and Ross Taylor made a patient half-century as the pair put on 131 for the third wicket.
But they slipped to 219 for five and required some late hitting from Corey Anderson, Daniel Vettori and Tim Southee to see them across the line.
They eventually reached their target with three wickets and seven balls to spare to finish top of the group standings with a maximum 12 points.
Bangladesh were put in and were just 29 for two after 10 overs, Trent Boult bowling Imrul Kayes and having Tamim Iqbal caught at slip.
Mahmudullah was dropped on one, a slash at Southee bursting through Anderson's hands at slip and away for four.
Sarkar inside-edged for four in the same over but despite that uncertain beginning, the pair scored at five an over as they led their side towards a competitive total.
Sarkar was the early aggressor, moving to 30 from 19 balls with six fours as Bangladesh's fifty came up in the 13th over.
Mahmudullah hit Boult for three fours in an over and after a quiet few overs, he hit Mitchell McClenaghan for four to bring up his team's hundred.
Sarkar was first to his half-century, from 55 balls, but holed out to Anderson the ball before his partner followed him to 50. Mahmudullah had faced 63 balls and hit five fours.
The first over of the batting powerplay brought 13 runs and a wicket as Shakib Al Hasan hit Anderson for two fours before edging to Luke Ronchi to depart for 23.
Mahmudullah swept Vettori for the first six of the innings and another off McClenaghan took him into the eighties before Rahman hit three fours in one Southee over.
A two off McClenaghan took Mahmudullah to another century, following his 103 against England, and he immediately hit three successive fours.
Rahman hit Grant Elliott for six but holed out next ball and the all-rounder's final over saw two more maximums, the first helping Mahmudullah finish on 128 not out.
New Zealand's innings followed a similar pattern, Brendon McCullum and Kane Williamson falling in the fifth over with 33 on the board before Guptill and Taylor led the fightback.
Guptill was averaging 31 for the tournament and Taylor less than 18 but they found their feet, the opener leading the way as he reached 50 from 44 balls with five fours and two sixes.
Taylor's first boundary took 20 balls to arrive but he began to share the load with Guptill and reached 5,000 runs in one-day internationals.
At halfway New Zealand were well ahead of the rate at 148 for two and Guptill's hundred came up a couple of overs later, from 88 balls with 11 fours and two sixes. He reached only 105, though, before lofting Shakib to long-on.
Taylor reached an 82-ball half-century while Elliott was going well, a six and five fours taking him to 39 from 34 balls before he was caught in the deep.
When Taylor followed for 56 the pressure was building, but Anderson blasted 39 from 26 balls with three fours and three sixes.
Vettori lofted Nasir Hossain for six over the off side and Southee hit a six and a four off consecutive balls from Shakib to take New Zealand to 290 for seven, Shakib taking 4-55.