New Zealand bowled tightly and fielded with high intensity after winning the toss to restrict Bangladesh to 244 all out.
Tamim Iqbal and Soumya Sarkar shared an opening partnership of 45 before Sarkar was bowled by Henry for 25 and Tamim, on 24, skied a simple catch to mid-wicket.
Shakib Al Hasan and Mushfiqur Rahim added 50 for the third wicket but the stand ended when Mushfiqur was run out for 19 following a mix-up.
Shakib reached his fifty off 54 balls before, on 64, he edged medium-pacer Colin de Grandhomme to wicketkeeper Tom Latham and Mohammad Mithun went for 26.
New Zealand started their chase confidently as Martin Guptill made a breezy 25 and Colin Munro 24 to move them rapidly on to 55-2.
Williamson should have been dismissed for eight, but wicketkeeper Mushfiqur dislodged the bails with his arm to waste the opportunity.
Williamson and Taylor added 105 to put New Zealand in a dominant position before Williamson was caught at deep mid-wicket by Mosaddek Hossain off Mehedi Hasan.
Latham fell for nought and when Taylor was caught down the leg-side New Zealand nerves were jangling.
The target was still comfortably within range, but De Grandhomme (15) and Jimmy Neesham (25) also fell as the tension mounted.
Henry was bowled by Saufuddin with seven required but Santner drove the first ball of the 48th over to the extra cover boundary to get New Zealand over the line.
