Bangladesh then reached 103-2 at stumps in reply to New Zealand's 469 all out on the second day of the first Test in Chittagong.
After starting the day at 280 for five, the visitors slumped to 342 for nine before Watling's second test century and Boult's unbeaten half-century in Chittagong.
Left-arm paceman Boult then removed Bangladesh opener Tamim Iqbal for a duck with his second ball, before a 36-ball half century to 22-year-old Mominul Haque (77 not out) took his side through to 103 for two at the close.
The hosts had visions of bowling New Zealand out before lunch when Bruce Martin (1) and debutant Corey Anderson (1) were out within the first two overs of the day. Doug Bracewell (29) then combined with Watling in an eighth wicket stand of 57.
After Bracewell fell to Sohag Gazi debutant legspinner Ish Sodhi (1) quickly followed.
Watling played a typically gritty innings when he was reprieved after scoring four. Television replays showed paceman Rubel Hossain had overstepped when Watling was caught in the gully.
The late rally appeared to rattle Bangladesh, with Anamul Haque (3) following Tamin Iqbal back to the pavilion early in Bangladesh's first innings, lbw to Bracewell.
But Mominul Haque's unbeaten 77 anchored the innings and put the hosts in a solid position after two days.
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