The following steps helped me getting autocompletion for Rust/cargo working in Zsh:

1. Create folder to install the completion functions: mkdir ~/.zfunc

2. Add folder to fpath in .zshrc before the compinit function: FPATH="$HOME/.zfunc:${FPATH}"

3. Use rustup to generate the completions:

$ rustup completions zsh > ~/.zfunc/_rustup
$ rustup completions zsh cargo > ~/.zfunc/_cargo