In the other match, England finished a woeful campaign with a rain-affected, nine-wicket win over Afghanistan at the Sydney Cricket Ground.
At Hamilton, New Zealand, Bangladesh's Mahmudullah (128) scored his second consecutive century to set up a competitive 288-7 total. Opener Martin Guptill (105) provided the platform for New Zealand's chase, but the tournament co-hosts wobbled getting to the end.
Shakib Al Hasan picked up four wickets for 40 runs and troubled New Zealand's middle order, but Daniel Vettori (16 not out) and Tim Southee (unbeaten on 12) guided the hosts home with seven balls to spare.
"It was a bit of a topsy-turvy sort of game," Guptill said. "They were on top for quite a lot of it, then we fought back and they fought back again. It's just nice to get the win under the belt and show some fight towards the end."
At the SCG, England's batsmen scored quickly to chase down the revised target of 101 off 25 overs after the bowlers, led by Chris Jordan (2-13) and Ravi Bopara (2-31), restricted Afghanistan to 111-7 off 36.2 overs.
England was eliminated from quarterfinal contention on Monday in a 15-run loss to Bangladesh.
Ian Bell scored a composed 52 not out and combined for an opening 83-run stand with Alex Hales, before Hamid Hassan (1-17) had Hales caught behind in the 14th over for 37, including two sixes.
There are two matches on Saturday on the penultimate day of pool play — India takes on Zimbabwe at Auckland, and Australia plays Scotland at Hobart.
Defending champion India will aim to remain unbeaten. Captain Mahendra Singh Dhoni, who has led India to nine consecutive wins in World Cup matches — encompassing its run to the title in 2011 and its first five pool matches this year — has indicated India will maintain an unchanged lineup unless there's injuries.
Dhoni said India had no incentive to rest players as it headed into the quarterfinals, and would likely do so only if a player was in danger of breaking down.
"If everyone is fit and available for selection we'll field the best 11," he said. "Because there are already long breaks between matches, that much rest should be adequate."
India has bowled out all five of its opponents — Pakistan for 224, South Africa for 177, the United Arab Emirates for 101, the West Indies for 182 and Ireland for 259.
Australia is also keeping its changes to a minimum, with Pat Cummins replacing spinner Xavier Doherty in the only change to the lineup for the Pool A match against Scotland.
Scotland comes into the match on a 13-match losing streak at three winless World Cups, and has never beaten Australia, but captain Preston Mommsen said the mood in his camp was positive.
"We know we are up against it tomorrow again," he said. "It's something that as a group, we are looking to embrace the experience ... it's very exciting for us. We have to stay level, stay composed for as long as possible, and find a way into the game as early on as possible."