Jeet Raval was bowled for 33 by Kamrul Islam about 10 minutes before the scheduled end of play, but de Grandhomme came out and hit two of the first five balls he faced for six to signal's New Zealand's intentions to try and complete victory inside four days. He hit two more in succession to end the game.
Bangladesh, who were dismissed for 160 in their second innings in the first test in Wellington after scoring 595 for eight, were bundled out for 173 in their second innings on Monday, setting the hosts a victory target of 109.
"Today was probably our best day of cricket throughout the series which probably gave us the upper hand," New Zealand captain Kane Williamson said.
"A lot of credit to the bowlers and the way they bowled on a good surface was a fantastic effort and to then tick off the runs tonight was very nice."
Neil Wagner had sparked Bangladesh's collapse from 92 for three to 106 for seven, with the left-armer capturing three of the four wickets to fall, including two in one over.
Taskin Ahmed (33) and Kamrul(25 not out), however, produced a 51-run ninth-wicket partnership to ensure New Zealand would need to chase in excess of 100.
Tim Southee had earlier achieved a personal milestone when he had Shakib Al Hasan caught by de Grandhomme at backward point for eight to bring up 200 test match wickets.
The 28-year-old is now behind Richard Hadlee (431), Daniel Vettori (361), Chris Martin (233) and Chris Cairns (218) on the country's all-time list.
"I guess those things are nice to tick off on the way but it's nice to also sit back and enjoy another test win," said Southee, who was named man-of-the-match after figures of 8-142.
New Zealand had a first innings lead of 65 runs after Henry Nicholls had combined with the tail to take their side to 354 though the left-hander missed out on his maiden test century when he was bowled for 98.
"We had our moments but I think we lost our way a bit," said Bangladesh captain Tamim Iqbal.
"Bowling them out for 350 was a fantastic job ... but I think our batters ...needed to take more responsibility and score more runs and put them under pressure."
(Writing by Greg Stutchbury in Wellington; Editing by Amlan Chakraborty)
