How to run a script as root in Jenkins?

You must run the script using sudo:

sudo /path/to/script

But before you must allow jenkins to run the script in /etc/sudoers.

jenkins    ALL = NOPASSWD: /path/to/script

@Igor Chubin's answer is 100% correct, but never open sudoer file with a normal editor. always use visudo

just type

sudo visudo

this will take you to /etc/sudoers and upon saving it will make sure that there is no error in formatting.

if you make an error in sudoer file, you will lose sudo access, so always use visudo