Apple - How can I make Apple's keyboard shortcuts work in Chrome?

Chrome doesn't call Apple's spell checking code and instead wrote their own keyboard handling routines. Since this is how the program was made, there's not much of a workaround except to do your typing in a program that calls Apple's text input libraries and paste the results into Chrome. (Similar to how Dictionary lookup using Control-Command-D didn't work for several years on Chrome either until the code was updated to call Apple's SDK for that function).

You can see this Chromium bug might apply and also, that Chrome is in the process of removing it's own spell checker / autocorrect mechanism and shifting to rely on Apple's implementation: https://code.google.com/p/chromium/issues/list?q=label:Cr-UI-Browser-Spellcheck

This may not be a great solution if you have too many shortcuts to transfer, however it is nice, quick, and simple if your shortcut list is modest, or you don't mind having to manually enter and update them:

Auto Text Expander for Google Chrome

The extension also can do backup and import/export via JSON for the future.