The South African-born wicket-keeper compiled a career-best 103 as the Black Caps recovered from 282-7 to post 469 all out by tea on the second day at the Zahur Ahmed Chowdhury stadium.
Bangladesh were 103-2 in reply at close, following a breathtaking counter-attack by 22-year-old Mominul Haque, who smashed an unbeaten 77 off 71 balls with 13 boundaries.
Mominul shared an unbroken third-wicket stand of 95 with debutant Marshall Ayub (21 not out) after two wickets had fallen for eight runs by the fourth over.
Watling's century was the cornerstone of the brilliant resistence by New Zealand's tailenders that frustrated Bangladesh's bowlers.
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Number 11 Trent Boult also hit a personal best of 52 not out, helping Watling add a valuable 127 runs for the 10th-wicket on the newly-laid pitch that provided only slow turn.
"It was great fun batting with Trent out there," Watling said. "I thought he played extremely well. He waited for the loose ball and played some nice shots through the onside.