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.
The Black Caps lost nightwatchman 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.
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.
"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.
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.
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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