brew install caskroom/cask/brew-cask
brew tap caskroom/versions
brew cask install sublime-text
Weird how I will struggle with this for days, post on StackOverflow, then figure out my own answer in 20 seconds.
[edited to reflect that the package name is now just sublime-text, not sublime-text3]
[edit]: As of 2022-11-07, above is incorrect.
See taylorthurlow’s correct answer.