The dismissals of Tamim for 95 and Shakib Al Hasan for 20 just before the tea break put the Black Caps on top, with bad weather preventing play in the final session.
"Yes, the momentum was with us, but we can't control the weather," said New Zealand batsman Ross Taylor.
"[The match] is probably still pretty evenly poised and the rain has given our fast bowlers an extra break.
"When we do come out tomorrow hopefully, weather permitting, we can strike early and get them seven, if not eight, down before the second new ball," he said.
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Before the twin blows, Bangladesh had made steady progress after skipper Mushfiqur Rahim won the toss and elected to bat in overcast conditions at the Sher-e-Bangla National Stadium.
Tamim, who smashed 17 fours in his entertaining 153-ball knock, put on 76 with Mominul Haque (47) for the third wicket and 67 with Marshall Ayub (41) for the second.
Trent Boult gave New Zealand their first wicket when Anamul Haque (seven) miscued a pull shot and Williamson took the catch.