Just stumbled upon a new (to me) tool named Plink. Plink is a command-line interface to the PuTTY back ends and can for example be used when writing batch scripts in Windows.
For example the following one-liner could be used to start a script on a remote machine:
plink firstname.lastname@example.org -pw password /folder/script.sh
Plink can be downloaded from the regular PuTTY download page.