Is there a Node.js equivalent of Perl's WWW::Mechanize?

What you're looking for is a scriptable/headless browser. In decreasing order of popularity, the options are:

Google Puppeteer - "Node library which provides a high-level API to control Chrome or Chromium"
Zombie - more lightweight; "Insanely fast, headless full-stack testing using Node.js"
slimerjs - scriptable Firefox. No commits since March 2018.
mechanize-js, which lacks documentation and has far less traction than any of the above alternatives

Abandoned projects