New Zealand bowled Zimbabwe out for 295 to win the game by an innings and 117 runs, with left-arm seamer Neil Wagner finishing with eight wickets in the match at Queens Sports Club.
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The second and final Test gets under way at the same venue on Saturday.
After three days of brutal domination at the hands of the tourists, Williams rose from his sick bed to hit the fastest Test century by a Zimbabwean as he reached three figures in just 106 deliveries.
The left-hander had spent the past two days off the field due to illness as New Zealand racked up a first-innings lead of 412, and came to the crease early on the fourth morning with Zimbabwe tottering on 124 for six, still needing 288 runs to make the Black Caps bat again.
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Williams added 118 for the seventh wicket with captain Graeme Cremer, who made 33, and went on to make 119 before succumbing to the left-arm spin of Mitchell Santner.
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