After Taylor scored an unbeaten 173 and Watling made 107 in a New Zealand total of 576 for six declared, Boult struck three times in six deliveries to send Zimbabwe crashing to 17 for four in their second innings.
Craig Ervine's unbeaten 49 helped the hosts recover to 121 for five at the close of day three at Queens Sports Club, but Zimbabwe still need 291 runs to make New Zealand bat again in the match.
While the first two days of the game had been distinctly one-sided, the third was even more brutal as Taylor and Watling added 253 for the sixth wicket for New Zealand, who went on to claim four Zimbabwean wickets in the first four overs of their reply.
The Black Caps lost night watchman Ish Sodhi in the second over of the day as they resumed on 315 for four, but that simply allowed Taylor and Watling to get on with their productive alliance.
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The pair hardly offered the Zimbabweans a sniff in the four hours that they were together, as Taylor brought up his 14th Test hundred and Watling registered his sixth.
They were finally parted five overs after the tea break when Watling pulled a short delivery to deep square-leg, and the dismissal led Kane Williamson to declare with a first-innings lead of 412.
"He told us to just be positive and bat with a little bit of intent," Taylor said of the captain. "You still have to respect the bowlers.
"We could have easily batted on a little longer but it was a good sign of his to declare when he did."
The New Zealand bowlers wasted no time in getting stuck into their opponents, as Tim Southee had Hamilton Masakadza caught at slip in the first over and Boult went on the rampage.
The left-armer bowled Brian Chari in his first over, then dismissed Chamu Chibhabha and Prince Masvaure with consecutive deliveries in his second.
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Ervine and Sikandar Raza Butt responded with an entertaining counterattack that saw them put on 69 for the fifth wicket in just nine overs, but Neil Wagner ended the fun when he dismissed Raza for 37 with a short ball.
The Zimbabweans were left staring down the barrel of an innings defeat, as the debilitating effects of not playing a Test since November 2014 became increasingly clear.
"It's never easy when its stop-start like that, but you can't use that as a reason for the way we've played in this game," said Masakadza.
"In the second innings, the guys got some good balls. The first innings was really poor, a lot of soft dismissals, and that's why we are in the position we are in."
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