Williamson made 114 to enable the Black Caps to reach 280-5 after they won the toss and elected to bat on a newly-laid pitch at the Zahur Ahmed Chowdhury stadium.
New Zealand were coasting at 244-2 when left-arm spinner Abdur Razzak, recalled to the Test side after two years, grabbed two of the three wickets that fell after tea.
Williamson was leg-before to Shakib Al Hasan in the penultimate over, before Razzak dismissed skipper Brendon McCullum (21) off what turned out to be the final delivery of the day.
Fulton put on 57 for the first wicket with Hamish Rutherford (34) and 126 for the second with Williamson on a slow wicket that hampered stroke-making.
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Fulton, who needed 156 balls to reach his 50, celebrated his third half-century in Test cricket by pulling Marshall Ayub for a six over mid-wicket.
But the 34-year-old from Canterbury fell just before tea, hitting a short ball from part-time spinner Nasir Hossain to Mominul Haque in the covers.
"We did a lot of good work in the first 85 overs, so it was disappointing to lose those last two wickets.
"We still have capable batsmen to come in and we will look to get around 350 to 400. That will give our bowlers something to bowl at. I would expect the wicket to get lower and slower.