After months and months of struggle, I have found that the solution is a lot simpler than you could imagine:
Instead of autocomplete="off" use autocomplete="false" ;)
As simple as that, and it works like a charm in Google Chrome as well!
August 2019 update (credit to @JonEdiger in comments)
Note: lots of info online says the browsers now treat autocomplete='false' to be the same as autocomplete='off'. At least as of right this minute, it is preventing autocomplete for those three browsers.
Set it at form level and then for the inputs you want it off, set to some non-valid value like 'none':
