Reports of Musk laying off the platform's entire human rights team were "not, from my perspective, an encouraging start", Turk said in an open letter.
The United Nations High Commissioner for Human Rights said he was writing with "concern and apprehension about our digital public square and Twitter's role in it".
He warned against propagating hate speech and misinformation and highlighted the need to protect user privacy.
Musk, the richest person in the world, took control of the platform a week ago in a contentious deal.
After completing his mammoth $44 billion acquisition, Musk quickly set about dissolving Twitter's board and sacking its chief executive and top managers.
Twitter on Friday fired roughly half of its 7,500-strong workforce.
"Like all companies, Twitter needs to understand the harms associated with its platform and take steps to address them," wrote Turk.
"Respect for our shared human rights should set the guardrails for the platform's use and evolution. In short, I urge you to ensure human rights are central to the management of Twitter under your leadership."
Turk posted the open letter on Twitter, where he has more than 25,000 followers.
Turk, an Austrian longtime UN official who took up his post as the UN rights chief on October 17, spelt out some fundamental human rights principles, urging Musk to put them at the heart of Twitter's management going forward.
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