Nyauchi then had Rahmat Shah (zero) and Hashmatullah Shahidi (four) while Muzarabani had Munir Ahmad (one) and rival skipper Asghar Afghan (14) to leave Zimbabwe rocking at 47-6.
Nyuaci’s previous best of 3-69 was on his debut against Sri Lanka at Harare last year.
Earlier, resuming at 133-5, Zimbabwe added 117 more runs to gain crucial lead.
Williams, who resumed on 54 lost his overnight partner Ryan Burl for eight but then found an able partner in Regis Chakabva (44) as the duo added 75 for the seventh wicket.
Williams completed his third Test century, and second in successive Tests, soon after lunch with a single to square leg.
But he fell while flicking spinner Amir Hamza straight to mid-wicket.
Williams had 10 boundaries while Chakabva smashed a six and six boundaries. Left-arm spinner Hamza finished with career-best figures of 6-75 — his second five-for in Tests after his 5-74 against West Indies at Lucknow in 2019. The second Test starts on March 10, also in Abu Dhabi.